Premixed flames with heat losses

The material that has been presented in this section is of an approximate, physical nature and leaves open questions. For example, why is the value of a so large These questions may be addressed by detailed analyses that begin with the conservation equations in diflerential form. Analyses of this type are developed in the following sections. An expression for a will be derived in Section 8.2, and ignition analyses will be considered in Section 8.3. 

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]. 

Current computational capabilities enable two-dimensional formulation to be solved numerically [47]. Here we first address a model problem from the viewpoint of asymptotics. 

The first objective of the following analysis will be to compute the burning velocity for a simplified model of a plane, one-dimensional, non-adiabatic flame. Under certain conditions involving nonvanishing heat losses, it will be found that no solution for the burning velocity exists, thus indicating the presence of flammability limits. Under other conditions, two burning velocities will be obtained, one of which presumably corresponds to an unstable flame. 

We shall focus our attention on the steady-state temperature distribution, which is assumed to be determined by the equation 

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It is also well known that there exist different extinction modes in the presence of radiative heat loss (RHL) from the stretched premixed flame (e.g.. Refs. [8-13]). When RHL is included, the radiative flames can behave differently from the adiabatic ones, both qualitatively and quantitatively. Figure 6.3.1 shows the computed maximum flame temperature as a function of the stretch rate xfor lean counterflow methane/air flames of equivalence ratio (j) = 0.455, with and without RHL. The stretch rate in this case is defined as the negative maximum of the local axial-velocity gradient ahead of the thermal mixing layer. For the lean methane/air flames,... 

The process of steady flame propagation into a premixed system is depicted in Figure 4.12 for a moving control volume bounding the combustion region <5r. The heat loss in this case is only considered to the duct wall. With h as the convection heat transfer coefficient, the loss rate can be written as... 

The control volume analysis of the premixed flame of Section 4.5.4 can be used together with the analysis here in Section 9.9 for the diffusion flame to relate the two processes. We assume the kinetics is the same for each and given as in Equation (9.102). Since we are interested in extinction, it is reasonable to assume the heat loss from the flame to be by radiation from an optically thin flame of absorption coefficient, k ... 

Computational studies of partially premixed flames have also been reported [11-16], Authors have tried to explain the variation of NO emission indices with the level of partial premixing using one or more of the following residence time, flame stretch, radiation heat loss, and chemical mechanism-based arguments. However, a complete explanation of the NO emission has not been offered in the literature. 

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