IEM-model

The application of DQMOM to the closed composition PDF transport equation is described in detail by Fox (2003). If the IEM model is used to describe micromixing and a gradient-diffusivity model is used to describe the turbulent fluxes, the CFD model will have the form... 

Note that in order to close (1.16), the micromixing time must be related to the underlying flow field. Nevertheless, because the IEM model is formulated in aLagrangian framework, the chemical source term in (1.16) appears in closed form. This is not the case for the chemical source term in (1.17). 

If is far from 0.5 (i.e., non-equal-volume mixing), the IEM model yields poor predictions. Alternative models (e.g., the E-model of Baldyga and Bourne (1989)) that account for the evolution of p should be employed to model non-equal-volume mixing. 

Consider an inert (non-reacting) scalar

[Pg.33]

The first term on the right-hand side is the expected value of the scalar Laplacian conditioned on the scalars having values r//, 33 An example of the time evolution of the conditional scalar Laplacian, corresponding to the scalar PDF in Fig. 1.11, is plotted in Fig. 1.12 for an initially non-premixed inert-scalar field. The closure of the conditional scalar Laplacian is discussed in Chapter 6. For the time being, it suffices to note the similarity between (1.36) and the IEM model, (1.16). Indeed, the IEM model is a closure for the conditional scalar Laplacian, i.e.,... 

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

The IEM model is easily extended to treat multiple feed streams and time-dependent feed rates (Fox and Villermaux 1990b). 

The IEM model is a simple example of an age-based model. Other more complicated models that use the residence time distribution have also been developed by chemical-reaction engineers. For example, two models based on the mixing of fluid particles with different ages are shown in Fig. 5.15. Nevertheless, because it is impossible to map the age of a fluid particle onto a physical location in a general flow, age-based models cannot be used to predict the spatial distribution of the concentration fields inside a chemical reactor. Model validation is thus performed by comparing the predicted outlet concentrations with experimental data. 

In this manner, the non-relaxing property of the IEM model is avoided, and a(f, f) can be chosen such that the limiting mixture-fraction PDF is Gaussian. Indeed, from DNS it is known that the conditional diffusion for the mixture fraction has a non-linear form that varies with time (see Fig. 6.2). In the GIEM model, this behavior is modeled by... 

At first glance, an extension of the IEM model would appear also to predict the correct scalar covariance ... 

However, if the correlation matrix p is rank-deficient, but the scalar dissipation matrix is full rank, the IEM model cannot predict the increase in rank of p due to molecular diffusion. In other words, the last term on the right-hand side of (6.105), p. 278, due to the diffusion term in the FP model will not be present in the IEM model. The GIEM model violates the strong independence condition proposed by Pope (1983). However, since in binary mixing the scalar fields are correlated with the mixture fraction, it does satisfy the weak independence condition. The expected value on the left-hand side is with respect to the joint PDF (c, f x, t). 

Likewise, if (F) (0) lies well below the reaction zone, the IEM model will collapse all points outside the reaction zone towards the mean values without passing through the reaction zone, i.e., the flame will be quenched even when the local reaction rate is infinite.66 Such unphysical behavior is avoided with the EMST model (Subramaniam and Pope 1999). 

The IEM model80 has been widely employed in both chemical-reaction engineering (Villermaux and Devillon 1972) and computational combustion (Dopazo 1994) due (mainly) to its simple form. The IEM model assumes a linear relaxation of the scalar towards its mean value 81... 

Note that since the right-hand side depends only on a, the IEM model satisfies the strong independence condition proposed by Pope (1983). 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

Note that it is not necessary to use the IEM model for (rV2<(/ i//). For example, a non-linear expression could be employed (Pope and Ching 1993 Ching 1996 Warhaft 2000). However, the linear form of the IEM model greatly simplifies the determination of the model constants. 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

Like the IEM model, the FP model weakly satisfies property (iv). Likewise, property (v) can be built into the model for the joint scalar dissipation rates (Fox 1999), and the Sc dependence in property (vi) is included explicitly in the FP model. Thus, of the three molecular mixing models discussed so far, the FP model exhibits the greatest number of desirable properties provided suitable functional forms can be found for (e 0). 

Some of the models for the conditional diffusion presented in Section 6.6 can be used directly to close the right-hand side of (6.173). For example, the IEM model in (6.84) yields the Lagrangian IEM (LIEM) model. With the LIEM, the drift and diffusion coefficients... 

The term A2Pr is a direct result of employing the IEM model. If a different mixing model were used, then additional terms would result. For example, with the FP model the right-hand side would have the form /3 = A3PC + A2Pr + AsPra, where rd results from the diffusion term in the Fokker-Planck equation. 

For the IEM model, it is well known that for a homogeneous system (i.e., when pn and .) remain constant and the locations ((). ) move according to the rates rn. Using the matrices defined above, we can rewrite the linear system as... 

Thus, when c = 0, we find a = 0, so that a = 0 and b = Pr as required for the IEM model.3 The vector (3 will generally be non-zero for inhomogeneous systems. In this case, a will be non-zero, even when r = 0. 

The DQMOM results for the IEM model can be compared with the multi-environment presumed PDF models in Section 5.10. In particular, (5.374) on p. 226 can be compared with (B.44), and (5.375) can be compared with (B.45). First, we can note that for the IEM model G = 0 and Gs = a. Likewise, y AT"1 + pnSa 4>)n) = 7 .,z and = Ban. Of the four models introduced in Tables 5.1-5.5, only the symmetric two-environment model in Table 5.2 has G = 0 and yM(an) + pnSa = 7Zan. However, because the spurious dissipation terms only ensure that the mixture-fraction variance is correctly predicted, the symmetric two-environment model does not have Gs = a and Mg 1 = Ban. Thus, the covariance matrix is not predicted correctly, as it would be if (B.43) were used. We can thus conclude that the multi-environment presumed PDF models are incomplete in the sense that they do not control as many of the moments as possible for a given choice of -/Ve. 

The multi-variate DQMOM method, (B.43), ensures that the mixed moments used to determine the unknowns (an,b n,. .., b Ngn) are exactly reproduced for the IEM model in the absence of chemical reactions.11 As discussed earlier, for the homogeneous case (capn = 0) the solution to (B.43) is trivial (an = 0, b yn = 0) and exactly reproduces the IEM model for moments of arbitrary order. On the other hand, for inhomogeneous cases the IEM model will not be exactly reproduced. Thus, since many multi-variate PDFs exist for a given set of lower-order mixed moments, we cannot be assured that every choice of mixed moments used to solve (B.43) will lead to satisfactory results. 

Vandermonde matrices often arise when reconstructing PDFs from their moments, and efficient numerical methods exist for solving (B.51) (Press et al. 1992). Although Vandermonde matrices are notoriously ill-conditioned for large Ne, this will usually not be a problem in applications of the DQMOM-IEM model since it is most attractive (relative to Monte-Carlo methods) when Ne is small. 

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