Convective-diffusive zones

In the present analysis, the outer convective-diffusive zones flanking the reaction zone are treated in the Burke-Schumann limit with Lewis numbers unity. Lewis numbers different from unity are taken into account where reactions occur. These Lewis-number approximations are especially accurate for methane-air flames and would be appreciably poorer if hydrogen or higher hydrocarbons are the fuels. To achieve a formulation that is independent of the flame configuration, the mixture fraction is employed as the independent variable. The connection to physical coordinates is made through the so-called scalar dissipation rate. 

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. 

The flame structure that has been calculated here is illustrated schematically in Figure 5.4. There are two zones through which the temperature varies a thicker upstream zone in which the reaction rate is negligible to all algebraic orders in followed by a thinner reaction zone in which convection is negligible to the lowest order in The reaction occurs in the thin downstream zone because the temperature is too low for it to occur appreciably upstream where the reaction is occurring, the reactant concentration has been depleted by diffusion so that it is of order times the initial reactant concentration. In the formulation based on equation (42), where c(t) is sought, the convective-diffusive zone need not be mentioned... 

A quantitative estimate of the magnitude of the effects of relative diffusion of reactant and heat can be obtained on the basis of the structure of the convective-diffusive zone of Figure 5,4. In this zone 6 = 0, and solutions to equations (19) and (22) are r = and Y = respectively, where the conditions x = Y = 1 have been imposed at the reaction zone (c = 0). Since the fuel mass fraction is q(1 — Y), the expansion of the... 

The principal features of these flame-speed results are readily comprehensible physically from the previous discussion. Consider, in particular, the powers of p and of the Lewis numbers in equation (78). The extra 1 in ni + 2 + 1 for jS comes from the thinness of the reaction zone in comparison with the flame thickness, (5. The remaining powers for jS (nj and 2) and the Lewis numbers come from the rate expression w and reflect the fact that in the reaction zone the concentration of reactant i is of order Le0j8 times its initial concentration because of the concentration reduction through the convective-diffusive zone. For off-stoichiometric conditions, species 2 is not depleted appreciably, and therefore the rate-reduction factor (Le2/j ) does not appear in equation (79). 

There is an upstream convective-diffusive zone in which only convection and heat conduction occur. In the variable defined in equation (5-18), the solution in this zone is = y o and... 

The second way in which the derivation of equation (5-75) can fail is for the thickness of the reactive-diffusive zone in the gas to become comparable in size with the thickness of the convective-diffusive zone. This occurs if (T — 7])/T becomes of order which would be favored by low overall... 

The theoretical results that have been discussed here allow k to be of order unity, a condition that sometimes has been referred to as one of strong strain, although the term moderate strain seems better, with strong strain reserved for large values of k. Small values of k are identified as conditions of weak strain under these conditions the reaction sheet remains far to the reactant side of the stagnation point, and by integrating across the convective-diffusive zone, a formulation in terms of the location of the reaction sheet can be derived [102], like that discussed at the end of Section 9.5.1. By combining the approximations of weak strain and weak curvature, convenient approaches to analyses of wrinkled flames in turbulent flows can be obtained [38]. 

From equation (66) it is seen that as p becomes large the reaction term in equation (46) becomes very small ( exponentially small, since P appears inside the exponential) unless t is near unity. Hence for 1 — t of order unity, there is a zone in which the reaction rate is negligible and in which the convective and diffusive terms in equation (46) must be in balance. In describing this convective-diffusive zone, an outer expansion of the form T = To(0 + Hi(P)ti 0 + H2(P)t2(0 + may be introduced, following the formalism of matched asymptotic expansions [35]. Here... 

The essential physical ideas apparently were first known to Zel dovich. The instability mechanisms can be explained by reference to Figure 9.9, where the arrows indicate directions of net fluxes of heat and species. A key element in the explanation is the fact that for large values of the Zel dovich number, the burning velocity depends mainly on the local flame temperature. This intuitively clear result is not entirely obvious from the formulation of Section 9.5.1 but may be motivated from it by employing RudYild = d Yi/d as a rough approximation to equation (95) in the preheat (convective-diffusive) zone for i = 1 if this equation is integrated with Ru constant and with the boundary conditions that Fj = ... 

Fig. 11.1 Diffusion flame structure determined on basis of one-step kinetics in an asymptotic approximation to the kinetic activation I - reaction zone II - convection-diffusion zone Fp - fuel concentration Fq - oxidizer concentration Daoc - the Damkohler number corresponds to an infinitely fast reaction Daext - the Damkohler number when flame extinction takes place...

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