Cascades stage separation factor

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... 

This relation between the heads and tails separation factors and the stage separation factor is the key property of an ideal cascade. 

To compare equilibrium times evaluated by approximate Eq. (12.204) and the lower bound Eq. (12.209), the example of an ideal cascade to perform the separation of Table 12.8 will be considered. It is assumed, in addition, that the stage holdup time h is 1 s and the stage separation factor is 1.0043, the nominal value for separating from UF6. For this... 

Figure 12.27 represents one stage of an ideal, close-separation, one-up, one-down cascade whose feed flows at rate 2M and contains Zj fraction U, Zg fraction U, and Zg = 1 — Zs — z fraction At the cut of used in such a cascade, heads flows at rate M and contains fraction and>>6 Stage tails flows at rateAf and containsXg fraction andxg Stage separation factors are defined as... 

It is proposed that B be concentrated by the gaseous diffusion process applied to BFs and "BF3. The plant is to be designed as an ideal cascade and is to separate feed containing 19 percent °B into product containing 90 percent and tails containing 8 percent. The stage-separation factor is 1.0074. 

An electrolytic hydrogen plant fed with natural water containing 0.0149 percent deuterium is operated as a simple cascade, without recycle, to produce water containing 0.142 percent deuterium. The stage separation factor a has the value 7.0, and the heads separation factor 0 is independent of stage number. 

This section is concluded with brief remarks on the time required to bring the cascade from start to first withdrawal of product. Due to their large size, many isotope separation plants have inventories that correspond to many days of normal production. O Equation (51.7) is an approximate relation between the time to first production, tp, feed and product isotopic compositions at equilibrium, single-stage separation factor, a, and the residence time per stage, h, for a cascade like that in O Fig. 51.3. The behavior of tp is similar to that for plant size, i.e., diverges to infinity proportionally to 1/(1 — a). ... 

Sections 0 and 0 demonstrate that almost any desired isotope can be separated by thermal diffusion or electromagnetic methods on a small scale. For large-scale separation, enormous amounts of material must be processed and energy consumption, cascade design, and other such factors become important. Tables 51.3 and 51.4 list theoretical upper limits of single-stage separation factors for H/D, and for several different separa-... 

In contrast, the actual separation factor using alumina membranes is only 1.0030 [Isomura, et al., 1969]. In practice, up to thousands of membrane tubes are arranged in a counter-current cascade configuration to achieve the required degree of separation for example, over 1,200 stages required for 3% and over 4.000 stages for 97% even with gas recirculation. 

The results for the isotopic separation of Na/ Na which were obtained by Knochel and Wilken in the system Dowex 50/aqueous or methanolic solution of cryptands are summarized in Table 13 (explanation for Krl and Kr see Chap. 4.3.1.2). To reach a high total enrichment compared with one equilibrium stage, the batch experiments were carried out as a cascade (Chap. 2.5.2). Then Eq. (20) was used for the calculation of a-values. To determine the isotopic separation factor Mr for the complex formation as well, the Kr-vuIucs were analyzed in the same system without cryptands " .iss) see Chap. 4.3.1.2). In all experiments 30 mg cation exchanger resin (Li - or Cs -form) were equilibrated with a 10M Na -solution where a lithium or cesium salt, which corresponds to the counterion of the resin, was added up to a total cation concentration of 10 M. If one has used a complexing ageftt, the initial cryptand concentration has been established to be 10 M (pH = 8). For most of the systems, the standard deviations given in Table 13 correspond to seven parallel experiments. The measurement of the radionuclides Na and Na was carried out as described in Chap. 4.2.4. 

Such a simple cascade, with an infinite number of stages each performing an infinitesimal amount of separation, is equivalent to type A differential st e separation. Equation (12.51) is equivalent to the form of the Rayleigh equation (12.35), when one recognizes that w in the simple cascade is equivalent to the heads separation factor 0 in differential stage enrichment, and a in the simple cascade is equivalent to the local separation factor a. ... 

If the separation factor for the system is known and the variation of the reflux ratio is specified as a function of stage number in the cascade, the number of ideal stages required to separate feed into product and tails of specified composition can be calculated. For example, starting with the known tails composition xn, the heads composition from stage 1,, is... 

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... 

Design example. The foregoing equations will be applied to the two>up, one-down ideal cascade considered by Oiander [01] having three stripping stages (tig = 3), seven total stages ( = 7), and a tails separation factor (7) of 1.3027. Values of r, s, and t then are... 

The stage holdup time h and separation factor a of a solvent extraction column for uranium enrichment are = 10 s, a = 1.0010. What is the minimum equilibrium time of an ideal cascade fed with natural uranium, stripping to 0.2 w/o and enriching to 3 w/o product Repeat for 90 w/o product. 

If the heads separation factor p is constant throughout such a simple cascade, the fraction of deuterium that may be recovered depends on the number of stages n and the overall enrichment oj in accordance with... 

A high, but attainable, overall separation factor of a = 7 will be used. A heads separation factor such as the one that would be used in an ideal recycle cascade of 3 = y/a = y/f = 2.646 will be assumed. Then the number of stages n is given by... 

In Fig. 13.15, the first m—2 stages constitute a simple cascade, operated without recycle, with constant heads separation factor 0. The recovery of deuterium from a simple cascade of m—2 stages operated at constant 0 is... 

O Equation (51.5) shows fVin increases as the overall separation is increased and as the separation factor a gets closer and closer to unity. For the latter reason, the minimum number of stages required for isotope separation can be very large. To cite a practical example, in a diffusion plant with a = 1.0043, producing 90% enriched product, and rejecting 0.3% tads, from natural uranium (0.7%) feedstock, w 1,870. Of course, under the conditions oftotal reflux no product is withdrawn, it is all held in the cascade as inventory. For finite product withdrawal, even more stages will be required. 

This is a respectably high separation factor, but, as shown in Practice Problem 1.6, it leads to a single centrifuge enrichment of only 10%. Because nuclear power plants require to be enriched from its natural abundance level of 0.7% to about 3.5%, considerable staging would still be required. Such staging is achieved in so-called countercurrent "cascades," an example of which is shown in Figure 1.5b. Much more about cascades and their crucial role in separation processes will appear in Chapter 7. 

Figure 9.1.1. (a) Countercurrent cascade of double-entry separating elements as in distillation (e.g). (b) Countercurrent cascade of singleentry separating elements, (c) Compositions of various streams for stages (n - Ij, n and (n -y 1) and separation factors. 

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