Well-mixed system

Applicability Neutralization would be appropriate for acidic and basic wastes. The process should be performed in a well-mixed system. Care should be taken to ensure compatibility of the waste and treatment chemicals to prevent the formation of more toxic compounds. 

Mixing plays an important role in digester operation Without well-mixed systems, the proeesses eannot aeeeptable levels of effieieney. There are a number of methods or eombination of methods whereby proper mixing is attained. These inelude ... 

This equation describes the change of population in a well-mixed system and is often used to model fully mixed crystallization and precipitation processes. If the system is imperfectly mixed, however, then the more complicated equation 2.88 can be used provided that the external flow field can be calculated e.g. by use of CFD (see later). 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

Analytical approaches to obtain the agglomerate size distribution are possible for well-mixed systems and when the rate of aggregation of clusters is defined by simple functions. In general, irreversible aggregation in well-mixed systems is described by Smoluchowski s coagulation equation, which... 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Note that only one system, the one corresponding to constant capture radius clusters in chaotic flows, behaves as expected via mean field predictions. In general, the average cluster size grows fastest in the well-mixed system. However, in some cases the average cluster size in the regular flow grows faster than in the poorly mixed system. 

The fractal nature of the structures is also of interest. Because of the wide range of flow in the journal bearing, a distribution of fractal clusters is produced. When the area fraction of clusters is 0.02, the median fractal dimension of the clusters is dependent on the flow, similar to the study by Danielson et al. (1991). The median fractal dimension of clusters formed in the well-mixed system is 1.47, whereas the median fractal dimension of clusters formed in the poorly mixed case is 1.55. Furthermore, the range of fractal dimensions is higher in the well-mixed case. 

Rate of aggregation of compact clusters in well-mixed systems follows Smoluchowski s theory. 

Grows algebraically for compact structures linearly for constant capture radius clusters in a well-mixed system. 

Has the fastest rate of growth in well-mixed systems. 

The macroscopic mass action rate law, which holds for a well-mixed system on sufficiently long time scales, may be written... 

A well-mixed system would have a standard deviation that would approach zero. If the mixture is poorly mixed, the standard deviation will be relatively high. 

Figure 19.8 Diffusivity D and concentration C at wall boundary, (a) Schematic view of a wall boundary. Diffusivity drops abruptly from a very large value DB, which guarantees complete mixing in system B, to the much smaller value Da. The concentration penetrates into system A when time t grows. X(/2(

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

Imagine a well-mixed system characterized by the concentration of one or several chemicals i, C( (Fig. 21.3). The concentrations are influenced by inputs 7 by outputs Oh and by internal transformation processes which occur either between state variables (for instance, between C, and C ) or between other chemicals X, Y,... which are not part of the set of state variables. 

Often, to simplify the analysis, the gas is assumed to be well mixed or to flow as a plug with no diffusion. In the well-mixed system, no gradients exist, and the set of coupled partial differential equations becomes sets of coupled algebraic equations, which is an enormous simplification. In general, however, spatial variations must be considered. 

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