Activation-energy asymptotics

Equation (71) is a conservation equation for the mixture fraction, and equations (76), (77), and (79) relate the other variables to this quantity and Yp. To study the structure of the reaction sheet, we therefore need an additional conservation equation—for example, that for fuel. From equations (1-4), (1-8), (1-9) and (1-12) we find that, under the present assumptions, this additional equation can be written as 

In this one-step approximation, the empirical reaction orders rip and Hq have been introduced, as discussed in Section B.1.3, since the overall stoichiometry seldom provides the correct dependence of the rate on concentrations. For simplicity here we put rip = Hq = I, obtaining a qualitatively correct variation of the overall rate with the mass fractions of reactants. The overall activation energy is , and the overall frequency factor for the rate of fuel consumption is Ap = VpBT . 

Since T appears inside the exponential, it is convenient to employ T instead of Yp as the variable to be considered in addition to Z. From equations (71), (79), and (86) it is readily shown that 

We consider that equation (71) is solved first, prior to the study of equation (88). This may be done in principle if pD is independent of T and if changes in the solution T have a negligible influence on the fluid dynamics. Otherwise it is only a conceptual aid, and we cannot investigate directly the variation of T with space and time. We can, however, investigate the variation of T with Z and study the influence of finite-rate chemistry on this variation. This type of investigation is facilitated by introducing into (88) the variable Z as an independent variable, in a manner analogous to that of Crocco [185]. 

Consider an orthogonal coordinate system in which Z is one of the coordinates and the other two, x and y, are distances along surfaces of constant Z. Let u and v be velocity components in the x and y directions, respectively, in this new reference frame. When use is made of equation (71) it is found that a formal transformation of (88) into the new coordinate system yields 

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Development of activation energy asymptotics for the mathematical analysis of combustion phenomena. 

Mallard-Le Chatelier theory, Semenov assumed an ignition temperature, but by approximations eliminated it from the final equation to make the final result more useful. This approach is similar to what is now termed activation energy asymptotics. 

This term specifies the ratio SJS and has been determined explicitly by Linan and Williams [13] by the procedure they call activation energy asymptotics. Essentially, this is the technique used by Zeldovich, Frank-Kamenetskii, and Semenov [see Eq. (4.59)]. The analytical development of the asymptotic approach is not given here. For a discussion of the use of asymptotics, one should refer to the excellent books by Williams [12], Linan and Williams [13], and Zeldovich et al. [10]. Linan and Williams have called the term... 

With simplified chemical kinetics, perturbation methods are attractive for improving understanding and also for seeking quantitative comparisons with experimental results. Two types of perturbation approaches have been developed, Damkohler-number asymptotics and activation-energy asymptotics. In the former the ratio of a diffusion time to a reaction time, one of the similarity groups introduced by Damkohler [174], is treated as a large parameter, and in the latter the ratio of the energy of activation to the thermal... 

In activation-energy asymptotics, E/R T is a large parameter. Equation (89) then suggests that Wj will be largest near the maximum temperature and that Wjr will decrease rapidly, because of the Arrhenius factor, as T decreases appreciably below Therefore values of the reaction rate some distance away from the stoichiometric surface (Z = ZJ will be very small in comparison with the values near Z = Z. This implies that to analyze the effect of w, it is helpful to stretch the Z variable in equation (92) about Z = Z, A stretching of this kind causes the terms involving the highest Z derivative to be dominant, and therefore equation (92) becomes, approximately. 

The advent of activation-energy asymptotics has helped greatly in clarifying criteria for the validity of the flame-sheet approximation. The... 

For steady-state diffusion flames with thin reaction sheets, it is evident that outside the reaction zone there must be a balance between diffusion and convection, since no other terms occur in the equation for species conservation. Thus these flames consist of convective-diffusive zones separated by thin reaction zones. Since the stretching needed to describe the reaction zone by activation-energy asymptotics increases the magnitude of the diffusion terms with respect to the (less highly differentiated) convection terms, in the first approximation these reaction zones maintain a balance between diffusion and reaction and may be more descriptively termed reactive-diffusive zones. Thus the Burke-Schumann flame consists of two convective-diffusive zones separated by a reactive-diffusive zone. 

FIGURE 5.4. Schematic illustration of deflagration structure obtained by activation-energy asymptotics. 

There have been a number of analyses employing activation-energy asymptotics for two-reactant, one-step chemistry. Most of these have been summarized by Mitani [40] reviews also may be found elsewhere [10], [13]. 

The earliest studies of heat-loss effects in premixed flames were based on analytical approximations to the solution of the equation for energy conservation [35] [39]. Two such approximations that have been sufficiently popular to be presented in books are those of Spalding (see [40]) and of von Karman (see [5]). Later work involved numerical integrations [41]-[43] and, more recently, activation-energy asymptotics [44]-[46]. 

In Section 5.3.6, activation-energy asymptotics have been applied to the adiabatic version of equation (9) for a particular rate function w burning-rate formulas are given in Section 5.3.6 for this rate function and in Section 5.3.7 for others. Here it is convenient to presume that for L = 0, the burning rate is known on the basis of these results and to employ the known adiabatic mass burning rate for the purpose of nondimensionalization. Thus, in analogy with equation (5-18), we introduce the nondimensional stream wise coordinate = m CpX/X and obtain the equation... 

It is attractive to seek solutions by use of activation-energy asymptotics. If there is a narrow reaction zone in the vicinity of x = 0, then by stretching the coordinate about x = 0 and excluding variations on a very short time scale, the augmented version of equation (56) becomes Xgd T/dx = — qgWi to the first approximation in the reaction zone. By use of equation (7-20), the integral of this equation across the reaction zone is seen to be expressible as... 

The necessary application of activation-energy asymptotics parallels that given in Sections 9.2, 5.3.6, and 8.2.1 and has been developed for both one-reactant [172], [173] and two-reactant [174], [175] systems. For most problems (for example, for stability analyses), gradients of the total enthalpy and of the mixture ratio of the reactants are negligible in the first approximation within the reaction sheet, so that in the scaled variables appropriate to the reaction zone, both 6 -h Y Q/iY qLq ) and Y -h y2Lej/(v2Le2) are constant. Evaluation of these constants is aided by the further result that Yi = 0 (and hence dYJd = 0) downstream from the reaction zone, at least in the first approximations, so that for the reaction-zone analysis, the expressions... 

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