Flame equations

Using this graph and the relationship it contains, one can now address the question of whether and under what conditions a laminar flame can exist in a turbulent flow. As before, if allowance is made for flame front curvature effects, a laminar flame can be considered stable to a disturbance of sufficiently short wavelength however, intense shear can lead to extinction. From solutions of the laminar flame equations in an imposed shear flow, Klimov [50] and Williams [51] showed that a conventional propagating flame may exist... 

This section concentrates on laminar premixed flames, which serve to illustrate many attributes of steady-state one-dimensional reacting systems. The governing equations themselves can be written directly from the more general systems derived in Chapter 3. Referring to the cylindrical-coordinate summary in Section 3.12.2, and retaining only the axial components, the one-dimensional flame equations reduce immediately to... 

The solution of the flame equations when the intermediates do not exist in their steady-state concentrations is an imposing task which is somewhat less forbidding after the simpler equations are studied. The single reaction flame where no intermediates are involved is still a substantial problem, but is quite tractable when the most ordinary of digital computers is available. After elimination of distance, as previously discussed, this set of two ordinary nonlinear lst-order differential equations is obtained ... 

In order to develop an intuition for the theory of flames it is helpful to be able to obtain analytical solutions to the flame equations. With such solutions, it is possible to show trends in the behavior of flame velocity and the profiles when activation energy, flame temperature, diffusion coefficients, or other parameters are varied. This is possible if one simplifies the kinetics so that an exact solution of the equation is obtained or if an approximate solution to the complete equations is determined. In recent years Boys and Corner (B4), Adams (Al), Wilde (W5), von K rman and Penner (V3), Spalding (S4), Hirschfelder (H2), de Sendagorta (Dl), and Rosen (Rl) have developed methods for approximating the solution to a single reaction flame. The approximations are usually based on the simplification of the set of two equations [(4) and (5)] into one equation by setting all of the diffusion coefficients equal to X/cpp. In this model, Xi becomes a linear function of temperature (the constant enthalpy case), and the following equation is obtained ... 

Validation of the Global Rates Expressions. In order to validate the global rate expressions employed in the model, temperature and concentration profiles determined by probing the flames on a flat flame burner were studied. Attention was concentrated on Flames B and C. The experimental profiles were smoothed, and the stable species net reaction rates were determined using the laminar flat-flame equation described in detail by Fristrom and Westenberg (3) and summarized in Reference (8). 

Let us now consider flames in which more than one chemical reaction occurs. If assumptions 1-7 of Section 5.2 and the isobaric approximation p constant are retained, then the flame equations may be taken as... 

The integral of equation (80) for i = r is then 6 = 0 because the initial (as well as final) chain-carrier concentration is zero. Since the production terms are functions of t and of the mole fractions relations such as equation (85) permit explicit representation of in terms of t and the remaining X., Thus if equation (85) is valid for all chain carriers, the mole fractions of all these reaction intermediaries may be eliminated from the flame equations, and the flux fractions of all of these species are zero. Since stoichiometry conditions relate the remaining 6 , only one independent expression remains in equation (80), and the problem is reduced to that of a one-step reaction. The flame equations may then be solved explicitly to give all the mole fractions, including the X, in terms of t. 

Special situations exist for which this procedure simplifies considerably. If the intermediary under consideration is not a chain carrier but is merely produced and consumed through unimportant side reactions, then the burning velocity and the composition profiles of all other species in the flame are virtually unaffected by the presence of this intermediary. The structure of the flame (excluding the X profile) can therefore be determined completely by setting = 0 in the flame equations. After this structure is determined, a, b and the coefficients of the linear differential operator Si X ) are known functions of t. Therefore, equation (90) reduces to a linear nonhomogeneous differential equation with known variable coefficients,... 

An equation of type (62) exists for each species present in the gas, and for the energy of the mixture. The time derivatives vanish in the stationary flame equations. 

Even in cases where the first method has been successfully applied, this can provide a stringent test for the accuracy of the derived data. A number of alternative methods for the numerical solution of the systems of flame equations associated with complex reaction mechanisms are now available [123,125-130]. 

Cyanogen has the linear structure 13.31 and the short C—C distance indicates considerable electron delocalization. It burns in air with a very hot, violet flame (equation 13.73), and resembles the halogens in that it is hydrolysed by alkali (equation 13.74) and undergoes thermal dissociation to CN at high temperatures. 

Detailed Kinetic Modeling. Recent advances in computation techniques (11) have made it much easier to compute concentration-distance profiles for flame species. The one-dimensional isobaric flame equations are solved via a steady state solution using finite difference expressions. An added simplification is that the energy equation can be replaced with the measured temperature profile. In the adaptive mesh algorithm, the equations are first solved on a relatively coarse grid. Then additional grid points could be included if necessary, and the previous solution interpolated onto the new mesh where it served as the initial solution estimate. This process was continued until several termination criteria were satisfied. 

In Eq. (1) V is the burning velocity (cm/sec), m is the eigen solution of the flame equations (gm/cm /sec), and p is the initial density of the combustible gases (gm/cm ). An elementary treatment of the flame equations for a unimolecular reaction A- B has been given by Hirschfelder (1959). More sophisticated treatments can be found in the works of Hirschfelder, Curtiss, Campbell, Penner, von Karman, Spalding, and many others. A bibliography and discussion of such studies is found in Evans (1952). 

Note that unlike (l)-(3) mass and momentum are not necessarily conserved across the flame. Equation (11) states that (locally) the net normal mass flux is proportional to the stretch factor k. The difference between the mass flowing in and out of a volume element of thickness 6, is therefore either accumulated inside the element and/or transferred (tangentially) along the flame surface. Equation (12) states that the tangential fluid velocity vector is not necessarily continuous at the flame front. 

It is convenient to express the flame equations entirely in terms of dimensionless variables and dimensionless groups. The reduced temperature is defined by... 

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