Cascades ideal

Equations 27 and 28 can be used in conjunction, along with the corresponding equations for the stripping section, to produce an ideal plant profile such as is shown in Figure 4 where F is plotted against for the example of an ideal cascade to produce one mol of uranium per unit time enriched to 90... 

Fig. 4. Characteristics of an ideal separation cascade for uranium isotope separations. For this cascade a = 1.0043 Nj- = 3484 stages AU = 153.08mol/t ...

Total Upflow in an Ideal Plant. The sum of the upflows from all of the stages in the ideal plant, or more simply, the total upflow, is the area enclosed by the cascade shown in Figure 4. An analytical expression for this quantity is obtained as the summation of all the stage upflows in the enriching section expressed as an integral ... 

For the ideal cascade, T is given by equation 28 and dx/dn by equation 26. Making these substitutions ... 

The great utility of the separative capacity concept Hes in the fact that if the separative capacity of a single separation element can be deterrnined, perhaps from equations 7 or 10, then the total number of such identical elements required in an ideal cascade to perform a desired separation job is simply the ratio of the separative capacity of the cascade to that of the element. The concept of an ideal plant is useful because moderate departures from ideaUty do not appreciably affect the results. For example, if the upflow in a cascade is everywhere a factor of m times the ideal upflow, the actual total upflow... 

The sum of the stage feed flow rates of all of the stages in an ideal cascade is just twice the total cascade upflow rate when (a — 1) is small with respect to unity, or... 

Figure 5 shows an ideally tapered enricher that has been replaced by three square cascade sections, a process called squaring-off the cascade (3—6). During the squaring-off process, two essential requirements must be kept in mind The interstage flow in all-square sections must always exceed the local value of E at all points in the cascade, and the squared-off cascade must contain a total number of stages which exceeds N. In order for the... 

Fig. 5. Design of a real cascade obtained by squaring off an ideal enriching section.

Each of the individual CSTR s that make up the cascade can be analyzed using the techniques and concepts developed in Section 8.3.1. The present section indicates how one may manipulate the key relations developed earlier to obtain equations that simplify the analysis of a cascade of ideal CSTR s. 

We begin by indicating a few generalizations that are relevant to the treatment of batteries of stirrled tank reactors. Consider the cascade of ideal CSTR s shown in Figure 8.10. For any individual reactor denoted by the subscript i the basic design equatidn developed earlier as equation 8.3.4 is appropriate ... 

Plots used in the graphical analysis of cascades of ideal CSTR s. 

Size Comparisons Between Cascades of Ideal Continuous Stirred Tank Reactors and Plug Flow Reactors. In this section the size requirements for CSTR cascades containing different numbers of identical reactors are compared with that for a plug flow reactor used to effect the same change in composition. 

The Stirred Tanks in Series Model Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal stirred tank reactors operating in series. The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor. 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

In principle, the forward and backward transfer rates can be computed directly from DNS (see Appendix A). However, they are more easily computed by assuming idealized forms for the scalar energy spectrum (Fox 1995). In the general formulation (Fox 1999), they include both a forward cascade (a) and backscatter (/() ... 

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