where Ymj is the rate of production of component m due to chemical reactions occurring in the jth environment. The parameter, E is the reciprocal of a characteristic micromixing time scale (see Section 5.1).

This approach provides a flexible yet simple framework for modeling turbulent reactive flow processes. When micromixing is fast, that is, when E is large, small-scale mixing will make the concentrations in all sub-groups equal. This will reduce the model to a cell balance model. When macromixing is fast, the volume fractions of different sub-groups are the same all over the reactor and the model will reduce to the generalized engulfment model. In such cases, if the number of sub-groups is specified as two, the model reduces to that of Baldyga and Bourne (1989). At intermediate levels of micro or macromixing, the model is able to simulate the interaction of mixing and chemical reactions. The model is sensitive to the initial volume ratio of the segregated reactants, which has been experimentally observed. Many other closure models (for example, Patterson, 1985; Li and Toor, 1986; Dutta and Tarbell, 1989) do not possess this important property. The model also satisfies two important limits: the slow reaction (pure mixing) limit and the fast reaction limit (Ranade and Bourne, 1991). It has been compared with other published models by Ranade (1993). It has also been used successfully for simulating reactive mixing of series-parallel reactions in semi-batch stirred reactors by Ranade and Bourne (1991). More recently, a similar, four-environment model was used by Kolhapure and Fox (1999) to simulate mixing in a sensitive LDPE reactor. These authors have shown good agreement with the results obtained using this approach and those obtained with full PDF simulations. The methodology of combining multi-environment models with a computational fluid dynamics framework has recently been formalized by Fox (1998). This approach looks quite promising and suitable for simulating liquid phase fast chemical reactions in turbulent flow processes.

There are a few other non-PDF approaches to simulating reactive flow processes (for example, the linear eddy model of Kerstein, 1991 and the conditional moment closure model of Bilger, 1993). These approaches are not discussed here as most of the engineering simulations of reactive flow processes can be achieved by the approaches discussed earlier. The discussion so far has been restricted to single-phase turbulent reactive flow processes. We now briefly consider modeling multiphase reactive flow processes.

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