Maximum mixedness

Maximum mixedness mixing occurs at the earliest possible point 

In a reactor with a segregated fluid, mixing between particles of fluid does not occur until the fluid leaves the reactor. The reactor exit is, of course, the latest possible point that mixing can occur, and any effect of mixing is postponed until after all reaction has taken place. We can also think of completely segregated flow as being in a state of minimiun mixedness. We now want to consider the other extreme, that of maximum mixedness consistent with a given residence-time distribution. 

The volume of fluid with a life expectancy between A and A + AX is 

Columns 3 through 5 are calculated from columns 1 and2 

The bounds on the conversion are found by calculating conversions under conditions of complete segregation and maximum mixedness. 

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Maximum Mixedness With a particular RTD, this pattern provides a lower limit to the attainable conversion. It is explained in Sec. 23. Some comparisons of conversions with different flow patterns are made in Fig. 23-14. Segregated conversion is easier to calculate and is often regarded as a somewhat plausible mechanism, so it is often the only one taken into account. 

Maximum mixedness Exists when any molecule that enters a vessel immediately becomes associated with those molecules with which it will eventually leave the vessel that is, with those molecules that have the same Bfe expectation. A state of MM is associated with every RTD. 

Empirical Equations Tabular (C,t) data are easier to use when put in the form of an algebraic equation. Then necessary integrals and derivatives can be formed most readily and accurately. The calculation of chemical conversions by such mechanisms as segregation, maximum mixedness, or dispersion also is easier with data in the form of equations. 

In contrast to segregated flow, in which the mixing occurs only after each sidestream leaves the vessel, under maximum mixedness mixing of all molecules having a certain period remaining in the vessel (the life expectation) occurs at the time of introduction of fresh material. These two mixing extremes—as late as possible and as soon as possible, both consistent with the same RTD—correspond to performance extremes of the vessel as a chemical reactor. 

Numerical solutions of the maximum mixedness and segregated flow equations for the Erlang model have been obtained by Novosad and Thyn (Coll Czech. Chem. Comm., 31,3,710-3,720 [1966]). A few comparisons are made in Fig. 23-14. In some ranges of the parameters n or fte ihe differences in conversion or reaclor sizes for the same conversions are substantial. On the basis of only an RTD for the flow pattern, perhaps only an average of the two calculated extreme performances is justifiable. 

Maximum mixedness model The fluid in a flow reactor that behaves as a micro fluid. Mixing of molecules of different ages occurs as early as possible. 

Maximum mixedness (Individual molecules are tree to move about and intermix). 

A real system must lie somewhere along a vertieal line in Figure 9-5. Performanee is within the upper and lower points on this line namely maximum mixedness and eomplete segregation. Equation 9-10 gives the eomplete segregation limit. The eomplete segregation model with side exits and the maximum mixedness model are diseussed next. 

The outlet eoneentration from a maximum mixedness reaetor is found by evaluating the solution to Equation 9-34 at = 0 =... 

When the residence time distribution is known, the uncertainty about reactor performance is greatly reduced. A real system must lie somewhere along a vertical line in Figure 15.14. The upper point on this line corresponds to maximum mixedness and usually provides one bound limit on reactor performance. Whether it is an upper or lower bound depends on the reaction mechanism. The lower point on the line corresponds to complete segregation and provides the opposite bound on reactor performance. The complete segregation limit can be calculated from Equation (15.48). The maximum mixedness limit is found by solving Zwietering s differential equation. ... 

Although this is an unusual solution to an ODE, it is the expected result since a stirred tank at maximum mixedness is a normal CSTR. 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

Equation (15.48) governs the performance of the completely segregated reactor, and Equation (15.49) governs the maximum mixedness reactor. These reactors represent extremes in the kind of mixing that can occur between molecules that have different ages. Do they also represent extremes of performance as measured by conversion or selectivity The bounding theorem provides a partial answer ... 

Suppose is a function of a alone and that neither dSt Ajda nor d Alda change sign over the range of concentrations encountered in the reactor. Then, for a system having a fixed residence time distribution. Equations (15.48) and (15.49) provide absolute bounds on the conversion of component A, the conversion in a real system necessarily falling within the bounds. If d S A/dc > 0, conversion is maximized by maximum mixedness and minimized by complete segregation. If d 0i A/da < 0, the converse is true. If cf- A/da = 0, micro-mixing has no effect on conversion. 

The difference between complete segregation and maximum mixedness is largest when the reactor is a stirred tank and is zero when the reactor is a PFR. Even for the stirred tank case, it has been difficult to find experimental evidence of segregation for single-phase reactions. Real CSTRs approximate perfect mixing when observed on the time and distance scales appropriate to industrial reactions, provided that the feed is premixed. Even with unmixed... 

Assume A, B, C, and D have similar diffusivities so that local stoichiometry is preserved. Under what circumstances will conversion be maximized by (a) complete segregation (b) by maximum mixedness Heterogeneous reactions are often modeled as if they were homogeneous. A frequently encountered rate expression is... 

Micromixing between these two extremes (partial segregation) is possible, but not considered here. A model for (1) is the segregated-flow model (SFM) and for (2) is the maximum-mixedness model (MMM) (Zwietering, 1959). We use these in reactor models in Chapter 20. 

In addition to these two macromixing reactor models, in this chapter, we also consider two micromixing reactor models for evaluating the performance of a reactor the segregated flow model (SFM), introduced in Chapters 13 to 16, and the maximum-mixedness model (MMM). These latter two models also require knowledge of the kinetics and of the global or macromixing behavior, as reflected in the RTD. 

The segregated-flow reactor model (SFM) represents the micromixing condition of complete segregation (no mixing) of fluid elements. As noted in Section 19.2, this is one extreme model of micromixing, the maximum-mixedness model being the other. 

The maximum-mixedness model (MMM) for a reactor represents the micromixing condition of complete dispersion, where fluid elements mix completely at the molecular level. The model is represented as a PFR with fluid (feed) entering continuously incrementally along the length of the reactor, as illustrated in Figure 20.1 (after Zwieter-ing, 1959). The introduction of feed incrementally in a PFR implies complete mixing... 

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

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