Interaction by exchange with

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... 

In a second kind of model, the interaction is represented via a deterministic process. Each particle is assumed to exchange matter with an average environment made up of all particles having the same life expectancy. For instance, the IEM Model (Interaction by Exchange with the Mean), was initially developed for representing micromixing in stirred reactors (33, 32). However, it can easily be extended to any RTD (84). The basic equation for conversion of a species in the L.E. is written... 

Interaction by Exchange with the Mean (lEM) model of David and Villermaux (1987). 

There are other more advanced micro-mixing models such as the engulfment, deformation, and diffusion (EDD) model [24-26] and the interaction by exchange with the mean (lEM) model [27], The reader should consult the original papers for details. 

Choose between different theories for modeling nonidealities in chemical reactors snch as residence time distribution (RTD), interaction by exchange with the mean (lEM), engulfment deformation (E), and probability density function (PDF). 

In the "Interaction by Exchange with the Mean" (EEM) model (Villermaux et al., 1972, 1986) it is assumed that both reactant flows are dispersed into lumps that exchange reactants with the bulk, that is considered to be equal to the "mean", so that reaction occurs in two zones (a, and b). 

Villermaux et al. (1972-1986) proposed a mass transfer model based on Interaction by Exchange with the Mean (lEM-model). The idea was that both reactant feed streams are broken up into volume elements (particles) that travel through the reaction mixture, and exchange reactants with it. So, A- particles absorb B and loose A. Reaction takes place in the particles. The peculiar characteristic of this model is that there is no separate "bulk". The environment of each particle has a composition that is put equal to the mean of the entire reactor. The reaction and the mass transfer are described by the following equations... 

We see that this spin polarization is a spherical oscillating function of distance. If the free electron approximation is good the kp can be expressed as (3ir n) and the effective electronic mass m = mo. The spin polarization interacts by exchange with a localized neighbour ion at distance Ra from the scattering centre. The corresponding energy is... 

Model of interaction by exchange with the mean 12.3.1. Principle... 

The model of interaction by exchange with the mean (lEM) is also a model based on the notion of EFPs and employing the PDFs of EFP populations. Unlike the CD model, the lEM model does not individually identify each EFP, but rather each class of EFPs. All EFPs iiutially having the same concentrations Ca and Cb will evolve in the same way under the lEM model, which was not the case with the CD model. Since the initial distribution P(Ca,Cb,/ = 0) of the population of EFPs is given, its subsequent evolution P(Ca,Cb,/) will be limited to displacing the lines in the histogram. 

The amount of iodide formed given by [I3-] + [I2] is a measure for the quality of mixing. Based on the reaction kinetics and the characteristic reaction time 1 2. Commenge and Falk [14] calculated the formation of iodine as a function of the mixing time in microstructured reactors. For the theoretical predictions, the authors used the relatively simple interaction by exchange with the mean (I EM) model described by ViHermaux [10]. The results obtained for different experimental... 

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