Reaction-progress vector conditional

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... 

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... 

The dimensionless vector of reaction-progress variables Y is then defined to be null in the initial and inlet conditions, and obeys... 

An ad hoc extension of the method presented above can be formulated for complex chemistry written in terms of yip and . In the absence of chemical reactions, y>rp = 0. Thus, if a second limiting case can be identified, interpolation parameters can be defined to be consistent with the unconditional means. In combusting flows, the obvious second limiting case is the equilibrium-chemistry limit where yip = y>eq( ) (see Section 5.4). The components of the conditional reacting-progress vector can then be approximated by (no summation is implied on a)... 

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