General formulation reaction-progress variables

Chemical reactions for which the rank of the reaction coefficient matrix T is equal to the number of reaction rate functions R, (i. e 1. I) (i.e., Nr = I), can be expressed in terms of / reaction-progress variables Y, (i. e 1./), in addition to the mixture-fraction vector . For these reactions, the chemical source terms for the reaction-progress variables can be found without resorting to SVD of T. Thus, in this sense, such chemical reactions are simple compared with the general case presented in Section 5.1. 

In order to simplify the discussion further, we will only consider the case where the molecular diffusivities of all chemical species are identical. We can then write the linear accumulation and transport terms as a linear operator  

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

the species concentrations can be expressed as a linear combination of Y, , and a constant vector Co 90 

Applying the linear operator to this expression yields 

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