Micromixing models

Another class of micromixing models is based on fluid environments (Nishimura and Matsubara 1970 Ritchie and Tobgy 1979 Mehta and Tarbell 1983a Mehta and Tarbell 1983b). The basic idea behind these models is to divide composition space into a small number of environments that interact due to micromixing. Thus, unlike zone models, which divide up physical space, each environment can be thought of as existing at a particular 

The IEM model for a non-premixed PFR employs two environments with probabilities pi and p2 = 1 - pi, where pi is the volume fraction of stream 1 at the reactor inlet. In the IEM model, pi is assumed to be constant.21 The concentration in environment n is denoted by f (n) and obeys 

By definition, averaging (1.16) with respect to the operator ( (defined below in (1.18)) causes the micromixing term to drop out 22 

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 a Lagrangian 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). 

The mean concentrations appearing in (1.16) are found by averaging with respect to 

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In what follows, both macromixing and micromixing models will be introduced and a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail. It will be used in Chapter 8 to model the influence of the hydrodynamics on a meso- and microscale on continuous and semibatch precipitation where using CFD, diffusive and convective mixing parameters in the reactor are determined. 

Baldyga, J. and Bourne, J.R., 1984c. A fluid mechanical approach to turbulent mixing and chemical reaction. Part III Computational and experimental results for the new micromixing model. Chemical Engineering Communications, 28, 259-281. 

Ritchie, B.W. and Togby, A.H., 1979. A three-environment micromixing model for chemical reactors with arbitrary separate feed streams. Chemical Engineering Journal, 17, 173. 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

Some typical results from their simulations are presented in Fig. 16 in which the yield XQ of the product Q from the slow reaction of a set of two competitive reactions in a fed batch reactor has been plotted vs. impeller speed for two micromixing models, viz. their own CSV model and Bourne s EDD model their simulation results are compared with experimental data from Bourne and Yu (1991). For the cases shown, the CSV model may perform better than Bourne s EDD model, in particular when A is fed near to the impeller where mixing is most intense. 

The CFD model developed above is an example of a moment closure. Unfortunately, when applied to reacting scalars such as those considered in Section III, moment closures for the chemical source term are not usually accurate (Fox, 2003). An alternative approach that yields the same moments can be formulated in terms of a presumed PDF method (Fox, 1998). Here we will consider only the simplest version of a multi-environment micromixing model. Readers interested in further details on other versions of the model can consult Wang and Fox (2004). 

Note that in the special case of size-independent growth, this term can be expressed as a closed function of the moments, i.e., G,t(c) = G(c)mk. Note also that when deriving Eq. (102) we have neglected the size-dependence of This is justified in turbulent flows and, in any case, to do otherwise would require a micromixing model that accounts for differential diffusion (Fox, 2003). 

Note that each environment in the micromixing model will have its own set of concentrations can and moments mkn, reflecting the fact that the PSD is coupled to the chemistry and will thus be different at every SGS point in the flow. The PD algorithm is applied separately in each environment to compute the weights (wmn) and abscissa (lmn) from the quadrature formula as follows ... 

We develop the CFD equations using the DQMOM model for micromixing. Nevertheless, care must also be taken when using other micromixing models, including transported PDF methods. 

In the second step we must add the micromixing terms from the DQMOM model to Eqs. (133)—(135). Fiowever, as we discussed earlier, we need to keep in mind that micromixing conserves the moments of the NDF, and not the weights and abscissas (see Eq. 113). The micromixing model in environment n for the bivariate moments has the form... 

From the discussion above, we should keep in mind that even if no SGS micromixing model is used to describe the multiphase flow, it may often be the case that chemical reactions (and indeed micromixing) will be limited by mass/ heat transfer between the phases. Because the multifluid model (see Eqs. 164 and... 

As a preliminary consideration for these two micromixing models, we may associate three time quantities with each element of fluid at any point in the reactor (Zwietering, 1959) its residence time, t, its age, ta, and its life expectancy in the reactor (i.e., time to reach the exit), te ... 

In the following sections, we first develop the two macromixing models, TIS and DPF, and then the two micromixing models, SFM and MMM. 

An important aspect of the micromixing models is that they define the maximum and minimum conversion possible for a given reaction and RTD. Zwietering (1959) showed that, for the reaction A - products, with power-law kinetics, ( -rA) = kAcA,... 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

Figure 1.6. Four micromixing models that have appeared in the literature. From top to bottom maximum-mixedness model minimum-mixedness model coalescence-redispersion model three-environment model.

In Chapter 6, this is shown to be a general physical requirement for all micromixing models, resulting from the fact that molecular diffusion in a closed system conserves mass. ( a)) is the mean concentration with respect to all fluid elements with age a. Thus, it is a conditional expected value. 

Choosing the micromixing time in a CRE micromixing model is therefore equivalent to choosing the scalar dissipation rate in a CFD model for scalar mixing. 

In the CRE literature, turbulence-based micromixing models have been proposed that set the micromixing time proportional to the Kolmogorov time scale ... 

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