The Well-Mixed System

Once we move away from single component systems there is the real possibility that the components will partition themselves in different parts of the vessel due to different densities, solubilities, or miscibilities. Partitioned systems are also referred to as distributed. That means that the properties are not everywhere the same over macroscopic length scales. To handle distributed systems we typically have to choose a differential control volume, that is, an infinitesimal volume within the macroscopic system. We will see this when we consider plug flow down a tube. 

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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. 

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) ... 

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. 

Differential equations governing the kinetics of chemical reaction systems may be thought of as arising from statements of mass conservation. For example, consider the well mixed system illustrated in Figure 3.1, containing reactants A and B in a dilute system of constant volume, V. 

Equation (139) shows that Cb/C will tend toward kj/kb as the material flow to the system becomes small, that is, as rCg and rC vanish. For r = 0, the system is a closed system, and Cb/Ca = kf/kb = K, die equilibrium constant. The quantity r = Q/V is the reciprocal of the fluid residence time of the well-mixed system r = trK As tr tends to very large values, the steady-sUite concentration ratio of the system approaches the equilibrium constant. 

The preceding theorem establishes that the well-mixed system being in the vicinity of a double-zero point is a necessary condition for a Turing bifurcation to occur in (10.1) with nearly equal diffusion coefficients. The next theorem establishes that this is also a sufficient condition. 

Theorem 10.3 If the well-mixed system is sufficiently close to a Takens-Bogdanov point, then there exists a set of dij such that a Turing bifurcation occurs for arbitrarily small e. 

We use the Brusselator, see Sect. 1.4.4, to illustrate these results. The steady state of the well-mixed system is stable ifb< = 1 + see (1.83). Condition (10.35) implies that... 

In order for the Turing bifureation to occur first, the Turing threshold must lie below the Hopf threshold of the well-mixed system, b - < bn-... 

Let p be a stable steady state of the well-mixed system ... 

The well-mixed system is operated with permeate pressure of 1.0 bar and retentate pressure of 4.5... 

The cross-flow system with unmixed permeate increases the separation compared to a conpletely mixed permeation system. The cross-flow system works better than the well-mixed system because the average driving force, [S(pj,yi.j - Ppyp j)l/N, is considerably larger in the cross-flow system than the single driving... 

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