Flame-sheet approximation

Since the flame-sheet approximation is common to all of the problems considered in Sections 3.1-3.3, it is of interest to study the character of this approximation in greater detail. The significance of the approximation can be addressed from the viewpoint of the general interface conservation condition given in equation (1-58). 

Criteria for the validity of the flame-sheet approximation may be developed by analyzing the structure of the sheet (see Section 3.4). For calculation of flame shapes in the Burke-Schumann problem, the approximation usually is well justified, although uncertainties arise for strongly sooting flames. 

In many respects, equation (57) is of more general validity than the burning rate expressions usually quoted because arbitrary temperature dependences of (pD) and Cp are permitted in equation (57), and the flame-sheet approximation has not been introduced. However, equation (57) will be useful for computing m only if all quantities appearing on the right-hand side of this equation are known. The specific heat at constant pressure for each pure species i(Cp ), the heat of vaporization per unit mass for the fuel at temperature Ti(L) and the quantity which is the heat of reaction... 

Since equation (42) is valid for any coupling function other results can be derived without explicitly invoking the flame-sheet approximation. For example, by considering... 

In the flame-sheet approximation, the position of the flame is determined by... 

FIGURE 3.6. Temperature and mass-fraction profiles for a burning fuel droplet in the flame-sheet approximation. 

If Le 1, then a formulation on the basis of coupling functions is less convenient. Equations (1-31), (1-32) and (1-33) become good starting equations for an analysis, especially if the flame-sheet approximation is adopted, since then the flux fractions remain constant on each side of the... 

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

FIGURE 10.3. Illustration of the dependences of the temperature and of the fuel and oxidizer mass fractions on the mixture fraction in the flame-sheet approximation, as given by equations (3-80H3-85). 

FIGURE 10.4. Illustration of probability-density functions for the mixture fraction, fuel mass fraction, oxidizer mass fraction, and temperature for a jet-type diffusion flame in the flame-sheet approximation. 

After Xc is obtained, the local average rates of fuel consumption and of heat release may be calculated, in the flame-sheet approximation, from the formulas... 

With F(rj) known, we may readily calculate u from equation (28) and pj from equation (47). Obtaining profiles in physical coordinates involves first evaluating p(/ )—for example, from pj and the ideal-gas law by introduction of a flame-sheet approximation—and then performing the integration in equation (27) for y. Profiles of the normal mass flux, pv, are given by equation (29), which—for the present problem—reduces to... 

VV( or q from equation (48) in the flame-sheet approximation. Since knowledge of the joint or conditioned functions is practically absent [27], statistical independence is often hypothesized or else it is merely assumed, less restric-tively, that the conditioned-mean dissipation equals the unconditioned mean. A small amount of data is available on unconditioned-average rates of scalar dissipation in turbulent flows (see discussions in [83]-[86]), and additional measurements are being made. These results allow estimates of Xc to be made, even though accurate calculations are beyond current capabilities. 

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