The Ideal Cascade

One type of tapered plant that is easy to treat theoretically, which has minimum interstage flow for a specified separation, and which is approximated by all isotope separation plants designed for minimum cost, is the so-called ideal cascade. An ideal cascade is one in which 

The heads stream and tails stream fed to each stage have the same composition  

Die theory of such cascades was developed by P. A. M. Dirac and R. Peierls in England and by 

Cohen and I. Kaplan in the Ihiited States and is described in The Theory of Isotope Separation by Cohen [C3]. The most important results are summarized in Secs. 8 throu 12 of this chapter, with some changes in terminology and notation. 

The above condition for an ideal cascade may also be expressed in terms of abundance ratios  

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

The overall separation depends on the single stage separation factor, the number of separating elements, and design and operating characteristics of the cascade. Equations 5-8 are insufficient to determine all the variables. It is instructive to consider three types of cascades the minimum stage cascade, the minimum reflux cascade, and the ideal cascade. The material balance equations from the i + 1 th stage to the product of the cascade lead to... 

The no remixing cascade is termed the ideal cascade. The ideal cascade retains the most desirable features of each of the two minimum cascades discussed and leads to a finite concentration in a finite number of stages at a finite production rate. For a cascade with a large number of stages... 

The ideal cascade requires twice the number of stages as the minimum stage cascade similarly the reflux ratio in the ideal cacade is twice that in the minimum reflux 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... 

Packing characteristics. We have shown that the optimum steam rate that leads to minimum tower volume and minimum power is that of the ideal cascade (13.14). The optimum typ>e of packing, optimum pressure, and optimum vapor velocity is that which makes the expression in braces (1327) a minimum. We shall not attempt to evaluate a number of types of packing, but shall use Spraypak no. 37 packing as an example of the selection of optimum vapor velocity and pressure. This is the type of packing recommended by McWilliams and co-workers [M4] for a water distillation plant. 

Figure 13.8 compares the variation of tails flow rate with stage number in a squared-off cascade with the variation in an ideal cascade performing the same job of separation in the same number of stages. Because the total flow rate in an ideal cascade is the lowest possible, the area under the stepped curve of the squared-off cascade is greater than under the smoothly tapered curve of the ideal cascade. 

Hydrogen from stage m — 1 constitutes tails from the ideal cascade section, whose quantity relative to product is given by... 

In Fig. 14.19, height above feed point is plotted vertically to correspond with orientation of an operating centrifuge. Optimum heavy-stream flow rate has a maximum of 0.2837 g UFj/s at the feed location (z = 0) and decreases to 0.0297 at the top and bottom. These are to be compared with 0.1884 g/s in the uniform-flow-rate case. This decrease in flow rate from feed location to withdrawal ends of the centrifuge is qualitatively similar to that of the ideal cascade discussed in Chap. 12. However, the tails flow rate at the top, product end of the centrifuge cannot drop to zero, as it would in an ideal cascade, because dyjdz would become zero at = 0, as can be seen from Eq. (14.250). 

Assume that the stage holdup time h, Eq. (12.198), is three times the above ratio. What is the minimum equilibrium time r, Eq. (12.209), of the ideal cascade of Prob. 14.1 ... 

The interstage flow rate L,- that must be maintained in order that there be any enrichm t in the ideal cascade at the point, where the concentraticm Xj occurs, is giv by... 

These basic results allow us to determine the number of stages needed in the stripping section and the enriching section to achieve a certain difference in the composition from the top product Xiu to the bottoms product Xi26 in the ideal cascade having a feed of composition xy. For the enriching section, we can start from stage 1 ... 

This problem is focused on the ideal cascade of Figure 9.1.1(h), where the enrichment achieved per stage is quite small, i.e. we have close separation. 

According to Eq. (15), the number of separating units in a cascade of specified total separative capacity U is minimized if the separative power of each unit is maximized. Consequently, we determine values of L, m, and 0 which maximize 6U. This optimization procedure is different from that discussed in Section II,H. In the latter, 9 and L were fixed and m was chosen to maximize a. In the present case, we shall fix m and choose 9 and L to maximize dU. Optimization of the controllable variables to give the largest separative power is more important than adjustments to achieve the largest separation factor dU determines the number of centrifuges required in a cascade, whereas a merely determines how the fixed number of units is to be arranged in the ideal cascade. 

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