Reaction-progress variables transport equation

By conditioning on the mixture fraction (i.e., on the event where f(x, t) = f), the reaction-progress-variable transport equation can be rewritten in terms of 7(f, r) 115... 

The microscopic transport equations for the reaction-progress variables can be found from the chemical species transport equations by generalizing the procedure used above for the acid-base reactions (Fox, 2003). If we assume that Fa Fb Fc, then the transport equations are given by... 

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 variable is defined such that 0 < Y < 1. However, unlike the mixture fraction, its value will depend on the reaction rate k = /> Bo. The solution to the reacting-flow problem then reduces to solving two transport equations ... 

The non-isothermal one-step reaction can then be described by a reaction-progress variable Y that is null when f = 0 or f = 1. As shown in Section 5.5, the transport equation for the reaction-progress variable is given by... 

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