Flame steady-state profile

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

The steady-state approximation is often used for the atomic and free radical intermediates occurring in combustion processes. The validity of this approximation has been examined in connection with the theoretical calculation of laminar flame velocities (3, 20, 21) in premixed gaseous systems. The steady-state approximation is occasionally useful for obtaining first-order estimates for flame-propagation velocities but should probably not be used in estimating concentration profiles for reaction intermediates. Some additional observations on the steady-state approximation are contained in Appendix I. 

Fig. 35. Computed quasi-steady state and partial equilibrium profiles for standard flame. Conditions as in Fig. 25. Solid lines, q.s.s. profiles broken lines, p.e. profiles (only marked when distinguishable from q.s.s.).

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 Z,. = 0 in the flame equations. After this structure is determined, a, b, and the coefficients of the linear differential operator S Xr) are known functions of t. Therefore, equation (90) reduces to a linear nonhomogeneous differential equation with known variable coefficients, and in the solution procedure described above S and b need not be recomputed (through the flame equations) in each step because they remain unchanged. For very small deviations from the steady state, it is often reasonable to omit the recomputation of S and b in successive steps even for chain carriers. 

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 the Eulerian steady-state, stationary flame the time derivatives in Eqs. (4.12) and (4.13) become zero, and the computational problem becomes one of determining convective fluxes and flame profiles that will allow this to occur everywhere. 

For slow flames, Eq. (4.79) may be uncoupled from the remainder of the calculation (as has been done so far), and Eq. (2.7b) may be used to determine the steady-state pressure profile at the end of the integration. For much faster flames where there are appreciable gasdynamic effects and associated density changes, the momentum equation must be coupled directly into the system, and the energy equations (2.19), (2.20) or (2.20q) must be used in place of Eq. (2.20b). In the finite-difference formulation discussed in Section 4.2, it then also becomes necessary to modify Eq. (4.44) to include the effect of variable pressure on the density and to introduce the condition... 

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