General micromixing model

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. 

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. 

Note that the E-model is asymmetric in the sense that environment 1 always grows in probability, while environment 2 always decreases. In general, this is not a desirable feature for a CFD-based micromixing model, and can be avoided by adding a probability flux from environment 1 to environment 2, or by using three environments and letting environment 2 represent pure fluid that mixes with environment 1 to form environment 3. Examples of these models are given in Tables 5.1-5.5 at the end of this section. 

As an example of a four-environment model,144 consider the generalized mixing model proposed by Villermaux and Falk (1994) shown in Fig. 5.23. The probability exchange rates r control the probability fluxes between environments.145 The micromixing terms for the probabilities can be expressed as... 

The general recycle model has been used by Weinstein and co-workers [55, 56] as a device for describing variable levels of micromixing. Dudukovic s work in this area has shown that approximations involved in Weinstein s work can introduce greater errors than were originally estimated [57] and he advocates the use of the more conventional micromixing models referred to in Sect. 4.3. 

Tsai, K. and Fox, R.O. (1995), Modeling multiple reactive scalar micromixing with the generalized lEM model, Phys. Fluids, 7, 2820. 

This equivalence also holds for predicting chemical conversion as can be seen in Fig. 12 where reaction extent was calculated for a second order reaction with unmixed feedstreams. The agreement is excellent. More generally, equivalence relationships can be established between all one-parameter micromixing models. For instance, the various models cited above yield approximately the same results under the equivalence conditions ... 

---