Models Explicitly Accounting for Mixing

Mixing limitations on the reaction rates can be quantified in terms of the Damkohler number(s)  

Data on the reactor behavior are most often only available at the reactor scale. This is no reason for modeling the reactor as a black box, however. Macro- and micro-scale phenomena affect the reactor scale behavior and the appropriate mixing details should be accounted for. 

Methods accounting for mixing are most easily illustrated for steady state or stationary reactor operation, as in Fig. 12.3-1. Because of its stochastic nature, turbulent flow is in fact only statistically iiormy. The random behavior of the variables results in rapid fluctuations of their values around mean or so-called Reynolds-averaged, steady state values. Nevertheless, turbulent flow is governed by deterministic equations, the Navier-Stokes equations, whose terms have been explained in Chapter 7 and in which a transient term is included to account for the fluctuations around the statistically steady state values. 

The Einstein notation is used throughout this chapter and a single reaction of A is considered. In (12.3-la), the species concentrations are expressed in terms of molar concentrations, C. An alternative form in terms of the species mass fractions, Y, is  

The conservation of mass is expressed by means of the total continuity 

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Macro- and Micro-Mixing in Reactors Models Explicitly Accounting for Mixing Micro-Probability Density Function Methods... 

Similar to RTD-based models (Section 12.6), multi-zone models do not explicitly account for the effects of micro-mixing. The latter may affect macromixing, that is, the transport rates between zones. Micro-mixing may also affect the averaged reaction rates at the scale of the zones or the macro-scale, as a result of non-uniformities within the individual zones. Micro-mixing effects can eventually be accounted for by the methods discussed in Sections 12.5 and 12.4. 

Using this approach, a model can be developed by considering the chemical potentials of the individual surfactant components. Here, we consider only the region where the adsorbed monolayer is "saturated" with surfactant (for example, at or above the cmc) and where no "bulk-like" water is present at the interface. Under these conditions the sum of the surface mole fractions of surfactant is assumed to equal unity. This approach diverges from standard treatments of adsorption at interfaces (see ref 28) in that the solvent is not explicitly Included in the treatment. While the "residual" solvent at the interface can clearly effect the surface free energy of the system, we now consider these effects to be accounted for in the standard chemical potentials at the surface and in the nonideal net interaction parameter in the mixed pseudo-phase. 

As mentioned above, the cross peak intensities from NOESY spectra taken at long mixing times caimot be related in a simple and direct way to distances between two protons due to spin diffusion effects that mask the actual proton distances. A possibiUty to extract such information is provided by relaxation matrix analysis that accounts for all dipolar interactions of a given proton and hence takes spin diffusion effects explicitly into consideration. Several computational procedures have been developed which iteratively back-calculate an experimental NOESY spectrum, starting from a certain molecular model that is altered in many cycles of the iteration process to fit best the experimental NOESY data. In each cycle, the calculated structures are refined by restrained molecular dynamics and free energy minimization [42,43]. 

This example shows that dipolar interactions can produce unexpected effects in systems containing polynuclear clusters, so that their complete quantitative description requires a model in which the dipolar interactions between all the paramagnetic sites of the system are explicitly taken into account. Local spin models of this kind can provide a description of the relative arrangement of the interacting centers at atomic resolution and have been worked out for systems containing [2Fe-2S] and [4Fe-4S] clusters (112, 192). In the latter case, an additional complication arises due to the delocalized character of the [Fe(III), Fe(II)] mixed-valence pair, so that the magnetic moments carried by the two sites A and B of Fig. 8B must be written... 

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