Description of Mixing

Finally, there is molecular diffusion or interpenetration of molecular species. It is responsible for the ultimate homogenization on a molecular scale (the ultimate particles are the molecules), and it is considered to be true mixing. This form of diffusion is driven by the chemical potential difference due to coneentration variation, and it is a very slow process, because its time scale is proportional to the value of the diffusion coefficient. Thus, this mechanism becomes important in gases and low-molecular-weight, miscible liquid systems, although there are time scale differences in those two cases. 

Some of the terms mentioned get a specific connotation when referred to polymer processing, and thus we give here some specific definitions (Matthews, 1982). Compounding refers to the process of softening, melting, and compaction of the polymer matrix and dispersion of the additive into 

FIGURE 6.2 Distributive mixing (a) random and (b) ordered rearrangements. (Reprinted by permission of the publisher from Tadmor and Gogos, 1979.) 

---

The bulk fluid velocity method relates a blending quaUty Chemscale number to a quaUtative description of mixing (Table 3). The value of is equal to one-sixth of the bulk fluid velocity defined by pumping rate divided by cross-sectional area of the tank (4). 

At the outset it is useful to consider some common examples of problems encountered in industrial mixing operations, since this will not only reveal the ubiquitous nature of the process, but will also provide an appreciation of some of the associated difficulties. Several attempts have been made to classify mixing problems and, for example, REAVELL(1) used as a criterion for mixing of powders, the flowability of the final product. HARNBY et at.(2) base their classification on the phases present that is liquid-liquid, liquid-solid and so on. This is probably the most useful description of mixing as it allows the adoption of a unified approach to the problems encountered in a range of industries. This approach is now followed here. 

Accurate description of mixing processes on each of these scales is only possible in a few selected and idealized cases. One of the best understood cases is that of a turbulent PBL over flat terrain and a point source of a trace substance. In this case, the concentration downwind of the source is often described as a plume. Figure 7-3 shows such an idealized plume. 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

Description of mixing with diffusion and reaction in terms of the concept of material interfaces. Journal of Fluid Mechanics 114, 83-103. 

The above concentration fluctuation information should aid in the fundamental description of mixing in packed beds and fluidized beds. Exactly how this information should be used in designing such systems must be the subject of further research. 

Scale of agitation Bulk fluid velocity (cm/sec) Description of mixing... 

Some complex standard models with many compartments may be simplified by using approximate age-dependent models with fewer parameters, and thus often with superior subsequent statistical analysis. One such application is the description of mixing in passage models. 

A useful description of mixing in bubble columns is provided by the dispersion model. The global mixing effects are generally characterized by the dispersion coefficients El and Eq of the two phases which are defined in analogy to Fick s law for diffusive transport. Dispersion in bubble columns has been the subject of many investigations which have recently been reviewed by Shah et al. (45). Particularly, plenty of data are available for liquid-phase dispersion. 

Li X., Chen G., Chen J., Simplified framework for description of mixing with chemical reactions. I Physical... 

An important intermediate level of description of mixing can be given in terms of the trajectories of fluid elements in the flow. This is the so-called Lagrangian description. Various characteristics of the ensemble of trajectories, like absolute and relative dispersion, contain useful information for predicting the evolution of the spatial distribution of quantities of interest. 

Moving up into the reactor level, effects of convection, dispersion and generation are described in the conservation equations for mass and energy. The momentum balance describes the behavior of pressure. The interface between the reactor and the catalyst level is described by the external mass transfer conditions, most often represented in a Fickian format, i.e., a linear dependence of the rate of mass transfer on the concentration gradient. In cases where an explicit description of mixing and hydrodynamic patterns is required, the simultaneous integration of the Navier-Stokes equations is also conducted at this level. I f the reaction proceeds thermally, the conversion of mass and the temperature effect as a result of it are described here as well. 

For the description of mixed monolayers, the choice of the dividing surface proposed by Lucassen-Reynders (see Eqs. 2.18, 2.19) is superior [58, 59]. The results obtained using the Butler equation (2.7) and Lucassen-Reynders dividing surface model for the description of mixed monolayers of non-ionic or ionic surfactants, and proteins assuming reorientation or aggregation of adsorbed molecules were presented and discussed in overviews [58, 59]. In this chapter, these concepts are discussed and further developed. 

Equivalent Circuit Description of Mixed Conduction in Solids... 

It will be convenient here to use the terms diffusive mixing for stage (c) and convective mixing for stages (a) and (b). It has not yet proved possible to develop a complete physical description of mixing in turbulent fluids. The present review will confine itself to simple models, which are supported by experimental work and which are (or could be) used industrially. 

In most mixing investigations it has been found that the prediction of power requirement can usually be dealt with satisfactorily by empirical relationships, usually based on dimensionless groups, derived from experimental results. The description of mixing rate is usually much more difTicult, as are the experiments, while scale-up has presented problems in most areas of mixing. For the static mixer the major variables requiring attention in any study are flow rate and viscosity, bearing in mind that in many processes there will be two or more fluids involved and that each inlet stream may have a different flow rate and viscosity. 

Despite wide use in commercial literature, there has been little application of the striation concept to experimental studies. Recent descriptions of mixing rate, both commercial and research, have been presented in terms of the reduction of variation coefllcient (a/C) with number of mixer elements or mixer length. It has been suggested that the variation coefllcient is more relevant to the description of commercial mixing processes than the relative standard deviation (a) or intensity of segregation ([Pg.236]

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

Thus we have strong experimental evidence that Flory-Huggins theory is inadequate as a quantitative description of mixing thermodynamics in polymer mixtures. From a theoretical point of view we can see four potential sources of error. 

For a more detailed description of mixing equipment see Wood [3]. 

The size of the particles of a clastic sedimentary rock allows it to be placed in one of three groups that are termed rudaceous or psephitic, arenaceous or psammitic and argillaceous or pelitic. Reference to size scales is made in Chapter 5, where a description of mixed aggregates also is provided. 

Description of mix 9.5 mm maximum-size aggregate 19 mm maximum-size aggregate 38 mm maximum-size aggregate... 

P., and Toye, D. (2014) CFD-based compartment model for description of mixing in bioreactors. Chem. Eng Scl,... 

---