Micro-mixing models

The macroscopic problem is more intricate. The type of model utilized depends upon the ratio of the diffusion and reaction rates and thus upon the importance of micro- and macro-mixing. In a pipe reactor the values of the axial dispersion coefficients for both phases are required. For modeling, micro-mixing models are used, which describe the mutual interlinking of coalescence and redispersion processes. 

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. 

The fundamental approaches to definition of turbulent flows macro-kinetics and macro-mixing processes are considered in [136-139]. Special attention was focused on micro-mixing models in the context of method based on equation for density of random variables probabilities distribution. Advantage of this method is that we can calculate average rate of chemical reaction if know the corresponding density of concentration and temperature possibility distribution. 

A somewhat less computationally demanding approach for calculating the composition field is based on the one-point joint composition PDF, f ), instead of the one-point joint velocity-composition PDF, f v,yf). With this approach, information on the turbulent flow / velocity field must be provided by appropriate flow, turbulence, scalar-flux and micro-mixing models. The reaction rate can still be exactly dealt with. A one-point joint composition PDF transport equation similar to the one-point joint velocity-composition PDF transport equation, (12.4.2-2), can be derived. For statistically stationary flow ... 

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

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). 

RATE CONSTANT BASED ON MICRO-MIXED CFSTR MODEL ... 

Calculated Molecular Weight Distributions. The calculated weight fraction distributions for the micro-mixed, segregated, and micro-mixed reactor with dead-polymer models for Runs 2, 5,... 

Evaluation of Mixing Models. The micro-mixed reactor will produce polymer disttibutions with increasing amounts of high molecular weight tail as the degree of polymerization of the polymer product increases over that of the original seed polymer. 

The micro-mixed reactor with dead-polymer model was developed to account for the large values of the polydispersity index observed experimentally. The effect of increasing the fraction of dead-polymer in the reactor feed while maintaining the same monomer conversion is to broaden the product polymer distribution and therefore to increase the polydispersity index. As illustrated in Table V, this model, with its adjustable parameter, can exactly match experiment average molecular weights and easily account for values of the polydispersity index significantly greater than 2. 

To differentiate between the micro-mixed reactor with dead-polymer and the by-pass reactor models in this investigation, the effect of mixing speed on the value of "( )" was observed. As illustrated in Table V, the value (j>" is not observed to increase with decreasing mixing speed as would be expected for a by-pass reactor. This rules out the possibility of a by-pass model and further substantiates the dead-polymer model. 

The micro-mixed reactor with dead-polymer model simulated the product of the laboratory reactor well within experimental accuracy. 

In spite of visual indications of at least partial segregation, the concept of micro-mixing proved to be most useful in modeling the laboratory reactor. 

From the viewpoint of dimensional analysis, the terms macro-mixing and micromixing used in the Theory of Turbulence are misleading, because they confuse the issue discussed above. In performing model experiments it does not matter whether the state of flow corresponds to the macro- or micro-mixing, but whether we succeeded in obtaining the working point of the same pi-space. 

The two extremes of the state of mixedness arc represented by the plug flow reactor (PFR, no mixing) and by the perfectly stirred reactor (PSR, perfectly mixed). The reactant flow in the PFR is neither macro nor micro mixed, whereas in the PSR mixing occurs down to the molecular level, thus both macro and micro mixing take place (see Figure 6). A variety of real flows can be characterised by series, parallel or loop connections of PFR and PSR. Additionally there exist other models such as the dispersion model (dispersed plug flow) which allows to model mixing conditions between the two extremes of PFR and PSR. 

For a given degree of macro-mixing, the degree of micro-mixing can vary. Both borderline conditions for micro-mixing can be demonstrated on the models of Danckwerts or of Zwietering, which both start out from an identical r distribution. 

In [16] a model was used, which combines micro-mixing and reaction with the disintegration of eddies by inertial forces. The structure of the model enables the disintegration of larger eddies by convective inertial forces to be quantitatively... 

The evaluation of diazotization reactions [41], which were carried out in Kenics and Sulzer SMXL mixers, provide a possible access to this parameter determination. For small throughputs and high viscosities the yield of the desired product was determined by micro-mixing. The power dissipation of 85-90% in both mixers indicated, that the engulfment model for micro-mixing prevailed. Faster micro- and meso-mixing was achieved in the Sulzer mixers, because larger pressure drops were also present in them, see Fig. 8.11 and 8.12. 

Bourne and coworkers [21-23] have developed the mathematics and applied their model to predict the effects of stoichiometric ratio, startup of a semi-batch reactor, effect of volumetric feed ratio, batch vs. continuous operation, etc. They have also experimentally demonstrated the use of reaction systems with well-characterized kinetics to determine the level of micro-mixing. Thus chemical reactions can be considered as molecular probes to be used to study segregation. Other... 

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