Chemical source term reaction-progress variables

Except for the chemical source term, these equations have the same form as those used for the mixture fraction. Note that the chemical source term (S oo) is evaluated using the mixture fraction and reaction-progress variable in the particular environment. The average chemical source term (S2oo(Y2) will thus not be equal to S2cc, (( ), (Y2)) unless micromixing occurs much faster than the second reaction. 

By definition of the reaction-progress variables, Y2 and T22 are zero for the inlet streams, and nonnegative inside the reactor due to the chemical source term. Once the CFD model has been solved, the reactant concentrations in each environment n are found from... 

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. 

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... 

Unlike for the mixture fraction, the initial and inlet conditions for the mean and variance of Y will be zero. Thus, the chemical source term will be responsible for the generation of non-zero values of the mean and variance of the reaction-progress variable inside the reactor. Appealing again to the assumption of independence, the mean reaction-progress variable is given by... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

Following the description given in the previous section, the detailed chemical system is mapped on two controlling variables the mixture fraction (Z) and a reaction progress variable (c). The mixture fraction describes the mixing of the species and enthalpy, while the progress variable follows the advance of the chemical reaction. Hence, the species mass fractions, temperature, density and species chemical source terms become functions of Z and c. The mixture fraction is defined according to [19] as ... 

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