Reaction-progress vector

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). 

In Section 5.5, we also introduce reaction-progress variables for simple chemistry that are defined differently than 5rp. Although their properties are otherwise quite similar, the reaction-progress variables are always nonnegative, which need not be the case for the reaction-progress vector. 

Figure 5.7. When the initial and inlet conditions admit a linear-mixture basis, the molar concentration vector c of length K can be partitioned by a linear transformation into three parts a reaction-progress vector of length NT , a mixture-fraction vector of length Nmf and 0, a null vector of length K — Nr — Nmf. The linear transformation matrix depends on the reference...

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... 

The solution to (5.151) as t oo is the local equilibrium reaction-progress vector 85... 

Finally, note that all of the expressions developed above for the composition vector also apply to the reaction-progress vector ip or the reaction-progress variables Y. Thus, in the following, we will develop closures using the form that is most appropriate for the chemistry under consideration. 

Since (5.299) is solved in mixture-fraction space, the independent variables are bounded by hyperplanes defined by pairs of axes and the hyperplane defined by X = K, = 1. At the vertices (i.e., V = = (0, and e, (/el,..., AW)), where e, is the Cartesian unit vector for the /th axis), the conditional mean reaction-progress vector is null 121... 

In other words, either die mixture-fraction PDF or die conditional reaction-progress vector (but not necessarily both) must be zero on the boundaries of mixture-fraction space. 

In a multi-environment conditional PDF model, it is assumed that the composition vector can be partitioned (as described in Section 5.3) into a reaction-progress vector y>rp and a mixture-fraction vector . The presumed conditional PDF for the reaction-progress vector then has the form 155... 

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 reaction-progress vector is premixed in the sense that variations due to finite-rate chemistry will occur along iso-clines of constant mixture fraction. 

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