Mixture-fraction vector transformation matrix

By definition of a and /3, 54 the sum of the Nm components of 7 is unity. Thus, by extending the definition of linear mixture to include the condition that (3 must be nonnegative,55 the last Nmf = Nm - I components of 7 can be used to define the mixture-fraction vector.56 The transformation matrix that links to ccv can be found by rewriting (5.84) using the fact that the components of 7 sum to unity ... 

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...

There are very few examples of scalar-mixing cases for which an explicit form for (e, 0) can be found using the known constraints. One of these is multi-stream mixing of inert scalars with equal molecular diffusivity. Indeed, for bounded scalars that can be transformed to a mixture-fraction vector, a shape matrix can be generated by using the surface normal vector n( ) mentioned above for property (ii). For the mixture-fraction vector, the faces of the allowable region are hyperplanes, and the surface normal vectors are particularly simple. For example, a two-dimensional mixture-fraction vector has three surface normal vectors ... 

Applying the same procedure to higher-dimensional mixture-fraction vectors yields expressions of the same form as (6.130). Note also that for any set of bounded scalars that can be linearly transformed to a mixture-fraction vector, (6.115) can be used to find the corresponding joint conditional scalar dissipation rate matrix starting from (e% C). 

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