Simple chemistry reaction-progress variables

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. 

For simple chemistry, a form for Q( x, t) can sometimes be found based on linear interpolation between two limiting cases. For example, for the one-step reaction discussed in Section 5.5, we have seen that the chemical source term can be rewritten in terms of a reaction-progress variable Y and the mixture fraction f. By taking the conditional expectation of (5.176) and applying (5.287), the chemical source term for the conditional reaction-progress variable can be found to be... 

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction . In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132... 

Various calculations of reacting flows, such as perfectly stirred reactors [12], laminar flames [13,14], turbulent flames [15,16], and hypersonic flows [17] have verified the approach presented above. Due to space limitation we shall only present one example, namely a premixed laminar flat flame calculation [13]. It provides a nice, simple test case for the verification of the model. The specific example is a syngas (40 Vol. % CO, 30 Vol. % H2, 30 Vol. % N2)-air system at p = 1 bar, and with a temperature of 290 K in the unburnt gas. The fuel/air ratio is 6/10. The influence of simplified transport models is described elsewhere [13]. Here, for the sake of simplicity, only systems with equal diffusivity shall be considered. In this case a three-dimensional manifold with enthalpy and two reaction progress variables as parameters has been calculated, i.e. the chemistry has been simpli-... 

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