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more than one reaction involving finite-rate chemical reactions, the application of moment closure or presumed PDF methods becomes increasingly unreliable. These methods are also not suitable for modeling reactive processes with multiple feed points. Several other inherent limitations of these methods are discussed in detail by Fox (1996).

One of the ways to overcome the difficulties associated with these methods is to use full PDF methods. These methods obtain the PDF by solving a balance equation for the one-point, joint velocity composition PDF wherein the chemical reaction terms are in closed form (Pope, 1981). Unlike presumed PDF methods, full PDF methods do not require a priori knowledge of the joint composition PDF, and the effect of chemical reactions is treated exactly. The one-point, joint velocity composition PDF or the one-point joint composition PDF does not contain information about fluctuations in the velocity and composition gradients (two-point information). Therefore, appropriate models are necessary to represent these terms and viscous and scalar dissipation in the joint PDF. It is possible to formulate a PDF approach using either a Lagrangian or Eulerian framework. Details of full PDF modeling are discussed by Pope (1985), Tsai and Fox (1995) and Fox (1996) among others, and will not be discussed here. The Lagrangian full PDF methods provide a link with phenomenological micromixing models (Fox, 1998). These phenomenological models may offer a computationally less demanding alternative to the full PDF methods and may adequately represent liquid phase reacting flows. These non-PDF closure approaches are discussed in the following section.

5.4.2. Phenomenological (non-PDF) Models

A number of simple, non-distributed models of reactive mixing have been developed. The engulfment model (E) of Baldyga and Bourne (1989) and the interaction by exchange with the mean (IEM) model of David and Villermaux (1975) are two examples of many such attempts. Non-distributed models (which mostly use a Lagrangian framework to describe local phenomena occurring in discrete fluid lumps) have been used successfully to simulate the interaction of micromixing and chemical reactions. None of these models, however, can account for the effects of bulk circulation, large-scale dispersion and local mixing that lead to spatial distribution of segregation, conversion and yield in the reactor. These models cannot, therefore, be used to simulate any general, three-dimensional reactive flow processes. It is necessary to develop an appropriate framework, which combines the advantages offered by these phenomenological models with the potential of using them in a general Eulerian modeling framework. Some of these attempts are discussed here.

(a) ESCIMO model. The acronym stands for the main constituent concepts of the approach, namely engulfment, stretching, coherence, inter-diffusion and moving observer (Spalding, 1978). In this model, mixing and chemical reactions occurring in small-scale coherent 'folds' are considered. The folds are formed by the engulfment of one fluid by another, as a consequence of 'roll up' of vortex sheets. The first part of the ESCIMO approach involves the solution of equations describing mixing and chemical reactions within these coherent 'folds' (biographical part). The second part (demographic part) involves determination of the composition of the population of these 'folds', within the reactor. With this approach, the mixing, molecular diffusion and chemical reactions occurring within each fold can be modeled rigorously, which is similar to any non-distributed, Lagrangian (moving observer) micromixing model. The second part poses significant difficulties because it requires representing three-dimensional reactive flow processes. It requires repeated solutions of transport equations defining the probability of finding a fold of any specific group at the specific location (Spalding, 1983). This restricts the applicability of the ESCIMO approach to simple flow processes, such as a well-stirred reactor. Ma et al. (1982) proposed the concept of 'limited-migration' to address this problem partially. Although the approach shows a promising way of combining Lagrangian and Eulerian viewpoints to simulate reactive flow processes, applications of this approach are rather restricted and not sufficiently validated by comparison with experimental data.

(b) Flamelet models. In this approach, the complex chemistry calculations are decoupled from the turbulent flow description by introducing the concept of coherent 'flamelet'. Multi-component transport and chemical reactions can be modeled rigorously for the flamelet. The results of local flamelet analysis can then be incorporated in the overall calculation of the turbulent flow field (Libby and Williams, 1980; Liew et al., 1984; Darabiha et al., 1989). For practical applications, a flamelet library (a database) is constructed to provide the required specific information, such as the consumption rates per unit flamelet area. The mean reaction rate sources required for the calculation of mean flow and composition fields are computed by taking the product of rate per flamelet area and flamelet area per unit volume. The latter quantity is obtained by solving its transport equation, which requires further modeling effort (Darabiha et al., 1989). In general, flamelet models are applicable to large Damkohler number systems (fast reactions) in which the typical turbulent scale is larger than the flame thickness. These models are therefore relevant to simulating IC engines and continuous flow combustors (at least for part of their operation). The flamelet approach has not been used to simulate liquid phase flow processes with fast reactions, for which multi-environment models may be more useful.

(c) Multi-environment models. In this approach, some of the micromixing models are extended to simulate interaction between macro- and micromixing by introducing the concept of multiple environments. Ritchie and Togby (1979) proposed a three-environment model; Mehta and Tarbell (1983) proposed a four-environment model. Ranade and Bourne (1991) have extended the engulfment model of Baldyga and Bourne (1989) to a general multi-environment model and have incorporated it in a Eulerian description of turbulent flow processes by developing transport equations for volume fractions of multi-environments. These models have the potential to simulate complex interactions between small-scale and large-scale reactive mixing. In the Ranade and Bourne (1991) model, the population of small-scale coherent fluid lumps (size of the order of the Kolmogorov length scale) is divided into N sub-groups (or environments). Each coherent fluid lump is assumed to have uniform concentration, implying that molecular diffusivity is not playing a significant role (a reasonable assumption for systems with Schmidt number <4000). The variations of concentration within these different sub-groups indicate incomplete micromixing and small-scale segregation. The large-scale mixing and transport of the small, coherent fluid lumps within the reactor (macromixing) is simulated using the general convective-dispersion transport equation in terms of general variable, 0:

â€” (pak 0k) + â€” (p Uiak 0k ) = â€” Y^akâ€”0k + (5.29)

where ak is the volume fraction of sub-group (or environment) k. When general variable, 0, is unity, the equation reduces to the governing equation of the volume fractions of different environments. For species concentrations, general variable, 0, is equal to Cmk , which is the concentration of the mth species in environment k. The species concentration, Cmk , may change due to convection, turbulent dispersion, micromix-ing and chemical reactions. The latter two terms are represented by the source terms in these equations, which are modeled using an extension of the engulfment model (Ranade and Bourne, 1991):

Saj = Eajaj+i + 2Eaj+i ^ ai - Eaj^ ai - Eaj ^ ai (5.30)

Scmj = Eajaj+1Cmj+1 + Eaj+1 ^ . ai (Cmi + Cmj+l) EajCmj ^ . ai i=1 i=1

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