Two-environment model

Many mixing models which utilize the simplified concepts of micro-mixing and segregation have been introduced. Most notable of these are the two-environment models of Chen and Fan (19), Kearns and Manning (20), and others (21, 22), and the dispersion models of Spielman and Levenspiel (23), and Kattan and Adler (24). 

The simplest model of this type is the two-environment model (N — 2) for which the independent state variables in the CFD model are... 

In theory, this model can be used to fix up to three moments of the mixture fraction (e.g., (c), ( 2), and (c3)). In practice, we want to choose the CFD transport equations such that the moments computed from Eqs. (34) and (35) are exactly the same as those found by solving Eqs. (28) and (29). An elegant mathematical procedure for forcing the moments to agree is the direct quadrature method of moments (DQMOM), and is described in detail in Fox (2003). For the two-environment model, the transport equations are... 

Although we will not do so here, with a little more work one can use Eqs. (36)—(38) to find the transport equation for ( 3). The two-environment model thus provides an extra piece of information that can be compared to experimental data. 

The next step would be to implement the CFD transport equation for the state variables in a CFD code. This is a little more difficult for the two-environment model (due to the gradient terms on the right-hand sides of Eqs. 37 and 38) than for the moment closure. Nevertheless, if done correctly both models will... 

The reader will recognize these terms as having of the same form as the correction terms in the two-environment model discussed earlier. With N — 1, 6 j = 0 and the model reduces to the laminar-chemistry approximation. With N —2, additional information is obtained concerning the second-order moments of the composition vector. Likewise, by using a larger N, the Mh-order moments are controlled by the DQMOM correction terms found from Eq. (89). 

In practice, it may be difficult to determine in advance which method is best to use for a particular application. For example, the CFD results may be more sensitive to large-scale inhomogeneities in the flow field than to the chemical source term closure. A rational approach to determine whether a more detailed SGS model is needed might be to start with N — 1 (laminar-chemistry approximation) and compare the predicted mean chemical species fields to the two-environment model (N — 2). If the differences are small, then the simpler model is adequate. However, if the differences are large, then the CFD simulation can be repeated with N — 3 and the results compared to N — 2. Naturally, once this procedure has converged, it will still be necessary to validate the CFD results with experimental data whenever possible. Indeed, it may be necessary to... 

Table 5.2. The terms in the transport equations for a symmetric two-environment model. 

Table 5.5.77 igrnw in the transport equations for a symmetric four-environment model that reduces to the symmetric two-environment model in the limit where p = p4 = 0. 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

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. 

Nishimura, Y. and M. Matsubara (1970). Micromixing theory via the two-environment model. Chemical Engineering Science 25, 1785-1797. 

It is useful to try to set some order in this host of models. For this purpose, the representation proposed by Spencer, Leshaw et al. (68) is helpful. It is essentially a two-environment model in which the assumption cited above Min. Mix. = macrofluid and Max. Mix. = microfluid is implicitly made. Along the axis of a small tube of the BPT model, the fluid gradually passes from a Min. Mix. state to a Max. Mix. state (Figure U). The residence time in this particular tube lies in the range tg, tg + dtg and the flow-rate is dQ = Q E(tg) dts. The flow transferred from the Entering Environment (E.E.) to the Leaving Environment (L.E.) in the interval... 

Figure 1. Two environment model for mixing, with schematic representations of the two extremes...

A generalization of these population balance methods to reactions with arbitrary RTD was given by Rattan and Adler [126]. They expanded the phase space of the distribution functions to include the life expectation as well as concentration of the individual fluid elements i/ (C, A, 0- The population balance then reduces to all of the previous developments for the various special cases of segregated or micromixed flow, the perfect macromixing coalescence-redispersion model, and can be solved as continuous functions or by discrete Monte Carlo techniques. Goto and Matsubara [127] have combined the coalescence and two-environment models into a general, but very complex, approach that incorporates much of the earlier work. 

Two-environment models in which one environment is in a state of complete segregation and the other in a state of maximum mixedness. 

A special kind of compartment model is the two-environment model, which divides the tank into micro- and macromixers, the numbers of which depend on the number of impellers. The flow behavior in the macromixer is characterized by the circulation-time distribution (Figure 3.3). Bajpai and Reuss [22] used a Monte Carlo simulation method in which the physical system of the macromixer was divided into a number of discrete elements. In each of these elements, the reaction process was simulated for a short period, at the end of which the system-specific interactions were simulated. The approach has been successfully applied to simulate the growth and metabolic overflow to ethanol at glucose concentrations beyond a threshold value for the yeast S. cerevisiae. 

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