Generalized Ideal Cascade

In the ideal cascade discussed up to this point, each stage receives as feed two streams of the same composition, a tails stream from the stage next higher in the cascade and a heads stream from the stage next lower in the cascade. In such a cascade the heads separation factor /, tails separation factor 7, and overall separation factor a are related by 

The cut thus ranges in value from l/(g + 1) at a = 0 to g/(g + 1) at z = 1. Because g for most isotope separation processes is close to unity, 6 in this type of ideal cascade must be close to 

In some isotope separation processes it is impractical to operate a stage at a cut of for mechanical or hydraulic reasons, and in others the separative capacity of the stage is higher at a cut substantially different from In the Becker separation nozzle process described in Chap. 14, the separative capacity of a stage producing a heads stream at a given rate is substantially hi er at a cut of than at a cut of j. 

To permit operation at a cut different from j while still ensuring that the composition of heads and tails streams entering each stage be equal requires a more complex cascade 

An extension of this development to the general, p-up, 7-down ideal cascade shows that the heads separation factor (I is 

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This holds for a close-separation, ideal cascade. When — 1 is not small relative to unity, the more general equation is... 

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 ... 

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 (/() ... 

A comparison of the various types of reactor concepts, in a general sense, is actually only possible between the batch, the CSTR and the PFR. The cascade of CSTRs, depending on the number of vessels n in the series, more or less behaves as an ideal mixer for n->l or an ideal plug flow for n- - . The fed-batch reactor is more difficult to situate. Although the concentration of compounds important for the rate of reaction can be controlled optimally during the whole fed period, the reactor volume is only partially utilized, especially in the beginning. Nevertheless, this reactor concept certainly has decisive advantages in many cases, as shown by the fact that it is one of the most widely used. 

Once a story is developed, it is crucial to stress test it. This can be achieved by looking at it from the viewpoint of the different audiences, such as customers, investors, or employees how would this particular audience respond to the telling of the current story What does it take to turn skeptics into believers and passive onlookers into active participants The next test would be to develop a full story variant for each audience. The variants for employees may be extended further in a story cascade . What form does the story take for key groups, such as business units, functional units, and front-line teams Is this a good story for the R D personnel, one that will increase the company s momentum in innovation Will the sales force be sufficiently excited and motivated by the story to reach the required sales volumes Of course, it is not really possible to define precisely what the story will mean to an audience, group, or particular individual. In the end, everyone connected with the corporation will tell their own version, but ideally in close alignment with the general theme. 

The above-mentioned measures do not combat the cause(s) or eliminate the primary pathogenetic reactions (with a few exceptions). The general aim is to prevent progression of the disease, ideally until the cirrhosis comes to a halt. When such medication is administered effectively, complicative developments can also be prevented. Pathogenetic primary reactions often initiate a cascade of secondary mechanisms, in particular of a biochemical nature. This is true, for example, of the inhibition of concomitant cholestasis during the course of cirrhosis by ursodeoxycholic acid (97), the reduction in lipid peroxidation by silymarin, the inhibition of fibrogenesis by colchicine (82) or silymarin (sometimes with improved quality of life), and the elimination of hyperammonaemia by ornithine aspartate. (145) (s. p. 279)... 

If the deviations are small then they can be described by the dispersion model (additional dispersive flow is is superimposed on the plug flow) or cell model (cascade of ideal stirred tanks). For larger deviations the calculation of nonideal reactors is generally difficult. A more simply treated special case occurs when the volume elements flowing through the reactor are macroscopically but not microscopically mixed (segregated flow). This case can be solved by the Hofmann-Schoenemann method (see below). 

Besides the dispersion model, the tanks in series model is the other onedimensional model widely used to represent non-ideal flow. Here the fluid is thought to flow through a series of equal-size ideal stirred tanks, and the parameter in this model is the number of tanks in the cascade (iV q). The RTD curves and moments of this model are easy to obtain, since problems of proper boundary conditions and method of tracer injection and measurement do not intrude. need not be an integer for curve-fitting purposes. It is strictly empirical, and no theoretical justification, such as Taylor diffusion, or theoretical estimates of the model parameter, are generally possible. This model starts from the mass balance equation for a series of i stirred vessels with 1 < i < iV, and AT, the number of vessels in the series (or the number of equivalent stages,... 

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