Mixedness, defined

The mixedness defined by Eq. (2.18) has values from zero for the no-mixing state to unity for the perfect-mixing state ... 

The distribution of the dispersed particle size is divided into m - 1 groups in the order of size, and each group is considered to be individual component. Additionally, the continuous phase is treated as another component. From this consideration, the mixing can be treated as m-component mixing, and the multi-component mixedness defined by Eq. (2.43) in the previous section can be applied. The extended definition of mixedness for the mixing of the continuous phase and dispersion phase can be expressed as... 

For n > 1, the segregated flow model provides the upper bound on conversion, and the maximum-mixedness model defines the lower bound. 

System Interaction rate State of segregation (here defined) Degree of segregation (Danckwerts) State of mixedness (Zwietering)... 

To clarify the relationship between the mixedness M defined by Eq. (2.18) and time t for the three typical impellers—disc turbine type, turbine type, and pitched type impellers. 

Nitrogen gas is fed at a fixed flow rate. After confirming that the flow in the column has a steady state and that the carbonic acid gas in the water is desorbed, the carbonic acid gas as the tracer is fed stepwise at the same flow rate as the fixed flow rate of nitrogen gas by changing the valve. The change in concentration of the tracer with time in each region is measured by making use of the electrode conductivity probe after the tracer gas is fed, and the mixedness M defined by Eq. (2.18) is calculated. 

Next, the case of m = n in Eq. (2.41), that is, the case that the number of components equals the number of imaginary equivolume partitioned regions, is considered. In this case, the denominator in Eq. (2.41) is log n and the formula coincides with the definition of the whole mixing capacity defined by Eq. (2.30). This is the point of contact between the multi-component mixedness and whole mixing capacity that is, both indices need not be discussed separately. 

It is possible to define an evaluation index for the mixing state by using the definition of multi-component mixedness in the previous section. The following discussion focuses on the mixing state of the continuous phase and the dispersed phase with a particle size distribution. 

However, the mixing of the dispersed and continuous phases is considered here, and it is possible to apply the same way of thinking for plural dispersed phases. Additionally, the newly defined mixedness can be applied to judge whether the assumption of MSMPR (mixed suspension mixed product removal) in the crystallization operation is established. 

To clarify the difference in mixedness M defined by Eq. (2.44) between the following two cases—the case where the two kinds of particles are treated as a particle (two phase mixing particle and continuous liquid phase) and... 

Here, the method of evaluating the separation state for an operation condition is discussed based on the consideration that the separation efficiency and mixedness should be related as the front and rear of an event. As given now, when an element of a substance in the vessel is selected, the separation state is evaluated from the viewpoint of information entropy based on the uncertainty regarding the component of the element that is selected. In order to define the separation efficiency that shows the degree of separation state quantitatively, the following conditions are set (Figure 3.2) ... 

The evaluation index of the separation operation/equipment is defined by using information entropy. This newly defined index has a clear quantitative relationship with the mixedness that was defined in Chapter 2. 

Win(ts,t) = 1 - F , fraction of material entering at time t which will remain in the reactor for a duration greater than tg, and Wout (ts,t) = 1 - Fout, fraction of material leaving at time t which remained in the reactor for a duration greater than tg. From these functions, two RTD can be defined, namely E n = 9F n/3tg and out = 3Fout/ats which have all the classical properties of steady state RTD except that they vary with time. In particular, chemical conversion can be calculated in the two limits of mixing earliness (see next Section). For minimal mixedness ... 

In addition to defining maximum mixedness discussed above, Zwietering also generalized a measure of micromixing proposed by Dankwerts and defined the degree of segregation, J, as... 

Clearly, at one extreme—when q(a) is zero throughout the reactor and we have a general j a)—we have the equations for a segregated-flow model. On the other extreme—when (a) is a Dirac delta exactly at one point and we have a general nonzero q oi)—this model reduces to the Zwietering (1959) model of maximum mixedness. Also, we define Q(a) as the flow of molecules at point a. Based on this nomenclature, a differential mass balance on an element Aa leads to... 

For layered or laminar mixtures, one can also define mixedness by the striation thickness, s (defined as 1/2 of the layer thickness). It can be shown that in these systems there is a simple relation Ay = 1/s. Since in real blends there is a variety of striation thicknesses, s should be expressed by a volume distribution function. 

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