Scalar vectors reaction-progress

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

Note that the reaction-progress vector in the first column is non-zero. Thus, as we suspected, the mixture-fraction basis is not a linear-mixture basis. The same conclusion will be drawn for all other mixture-fraction bases found starting from (5.118). For these initial and inlet conditions, a two-component mixture-fraction vector can be found however, it is of no practical interest since the number of conserved-variable scalars is equal to Nq,m = 1 (k e 0, 1, 2). In conclusion, although the mixture fraction can be defined for the... 

Having demonstrated the existence of a mixture-fraction vector for certain turbulent reacting flows, we can now turn to the question of how to treat the reacting scalars in the equilibrium-chemistry limit for such flows. Applying the linear transformation given in (5.107), the reaction-progress-vector transport equation becomes... 

As shown in Chapter 5, the composition vector can be decomposed into a reaction-progress vector tp and the mixture-fraction vector. Here we will denote the reacting scalars by [Pg.303]

The method just described for treating multiple reacting-progress variables has the distinct disadvantage that the upper bounds must be found a priori. For a complex reaction scheme, this may be unduly difficult, if not impossible. This fact, combined with the desire to include the correlations between the reacting scalars, has led to the development of even simpler methods based on a presumed joint PDF for the composition vector... 

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