### 7.8.1. Laminar Flamelet Model for Non Premixed Combustion

A diffusion flame is characterized by the diffusion of reactants into the flame front. While convective and diffusive time scales are of the same order of magnitude, the chemical time scales are much smaller for typical combustion processes of interest. Several approaches to treat chemical reactions have been developed and tested during the last decades.

The assumption of local chemical equilibrium has often been used in modeling the fast chemistry regime. For hydrocarbon flames, however, the assumption of local chemical equilibrium results in an over-prediction of intermediates like CO and H2. This suggests that non equilibrium effects are important in modeling these flames. Further essential non equilibrium effects are flame extinction, lift-off and blow-out.

Another well known approach is the flame sheet model of Burke and Schuman often characterized as 'mixed is burned'. Here, only the mixture of the reactants is calculated and the chemistry is treated as infinite fast and complete when mixing is complete. Therefore, combustion occurs in an infinitely thin sheet at the surface of stoichiometric mixture. Again non equilibrium effects are not taken into account, which are important if the strongly varying time scales of the turbulent flow fields approach those of the chemical reactions.

Linan [42] was the first who incorporated non equilibrium effects in diffusion flames. He analyzed the inner structure of the thin laminar flame sheet, referred to here as a Flamelet using an asymptotic description with a large Damköhler number as the expansion parameter. The Damköhler number is the ratio of flow to chemical time scales:

| (7–20)

Linan’s method is similar to Prandtl's boundary layer theory. The inner layer of the thin reaction sheet with well defined structure is called “Flamelet” from now on. A more simple description of flamelets is possible by using the mixture fraction, which is the sum of all elementary mass fractions:

| (7–21)

which have its origin in a system consisting of fuel inlet (labeled 1) and oxidizer inlet (labeled 2). Here is the mass fraction of species i, the mass fraction of a chemical element (such as C or H), the molecular mass, and the number of elements in the molecule

| (7–22)

Assuming equal diffusivities and heat capacities for all chemical components a conservation equation for the mixture fraction Z can be derived by summing all species conservation equations, and the chemical source terms therefore cancel exactly. The mixture fraction is not influenced by chemical reactions because it deals with elements rather than molecules, and elements are not affected by chemistry.

| (7–23)

The isosurface determines the location of stoichiometric mixture. To be able to describe the location of the flamelets anywhere in the flow field, a new coordinate system is introduced here. One of its coordinates is locally perpendicular to the surface of stoichiometric mixture.

The transformation is shown for the temperature equation as an example. The 3 terms at the right hand side represent chemical reactions, radiation and the transient pressure gradient, respectively. The last is important for combustion involving fast changing pressure such as in closed burning chambers of a piston engine.

| (7–24)

After applying the following transformation rules:

| (7–25)

| (7–26)

| (7–27)

and using the assumption of a constant Lewis number:

| (7–28)

you obtain the temperature equation in the form:

| (7–29)

Because the Flamelet is assumed to be thin, only gradients normal to the surface of stoichiometric mixture are large, and all terms without a second derivative in respect to the mixture fraction Z can be neglected. When this is done formally by introducing a stretched coordinate, it turns out that the remaining equation is essentially one dimensional. The same arguments apply for all other equations as well:

| (7–30)

Non equilibrium effects - the influence of the outer flow field on the inner reaction zone - are described by the scalar dissipation rate at stoichiometric mixture.

| (7–31)

It represents a reciprocal residence time that is increased by stretch effects of the flow field and reduced by diffusion. At a critical value of the flame shows a threshold behavior and extinguishes. The stretch in physical space leads to a reaction zone that is so thin that the production of heat cannot balance the heat loss caused by conduction. The temperature drops to unburnt values and the reactions freeze. `Freeze' means they are at lower temperatures and are much slower than the fluid time scales.

The important conclusion of this derivation is that flamelet structures in the presence of fast chemistry can be described by one dimensional model flames. This will be used to model reacting turbulent flow fields.

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