Cascade separation factors

B is a powerful neutron absorber and has been employed in reactor control rods, neutron detectors, and other applications. Cascades based on exchange distillation of boron-ether complexes have usefully large a s and were used for 10B/UB isotope separation by the US DOE. Exchange distillation takes advantage of the fact that condensed phase/vapor phase separation factors can be enhanced (as compared to liquid/vapor a s) by association/dissociation equilibria in one or the other phase. At the normal boiling point (173 K) the VPIE for... 

A schematic diagram for the enrichment of by gaseous diffusion of UFe through an effusion barrier is shown in Figure 5, which also illustrates the counter-current flow and cascade principles. The limiting separation factor a is given by the kinetic theory of gases... 

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 size of a separating element or the number of parallel elements necessary is proportional to the flow. The minimum reflux condition, Equations 18-20, shows that the area of the separation cascade or the amount of material to be processed at any point varies inversely with c. The minimum total size of the plant varies as c or (a — 1) There is a big premium on high separation factors. 

Our brief discussion of cascade principles serves to demonstrate the critical dependence of the size and operating costs of isotope separation plants on the elementary separation factor c. The size and initial cost are proportional to c 2. The operating cost is less sensitive to c, but varies at least as c The economic importance of these factors is readily seen in context with the separation of In 1960 the USAEC had three gaseous diffusion plants in operation. The cost of each plant was approximately 1 billion dollars the power consumption in each plant was 1,800,000 kw. If the plants were to be built with processes or equipment giving separation factors one half the one used, the additional construction cost to the U.S. taxpayers would be nine billion dollars. The increase in the annual operating costs of the plants can be conservatively estimated from the increase in the reflux ratio or power consumption to be 100,000,000/yr. This is a realistic demonstration of the economic benefits and importance of fundamental research and development to society. 

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. 

Often in practice, several membranes for gas separation are cascaded with recycle streams to effect a multistage system for increasing the separation factor to an effective level, particularly when the individual gases in the mixture have similar permeabilities. 

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. 

The separation factor in ail of these processes is so close to unity that production of separated isotopes requires repeated partial separations in a multistage cascade generally similar to the gaseous diffusion cascade of Fig. 12.2. The remainder of this chapter develops theoretical principles of isotope separation in such cascades. 

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... 

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... 

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... 

Figure 12.25 Compositions and separation factors in two-up, one-down ideal cascade.

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

The cascade receives feed of fraction Zp at flow rate F and produces an upper product of fraction yp at flow rate P, a lower product of fraction yg at flow rate Q, and tails of fraction Xw at flow rate W. For this two-up, one-down cascade, p = 2, q = I, the heads separation factor /3 is... 

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... 

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. 

A water distillation plant to produce heavy water containing 99.8 a/o D from natural water containing 0.015 percent D is designed as an ideal cascade without stripping with a separation factor of 1.03. The plant s inventory of water is effectively all in the liquid phase. The depth of liquid on each distillation plate is 30 cm. The plate efficiency is 100 percent. The liquid downflow rate is 1 cm /s per cm of column cross section. Using Eq. (12.204), what would be an upper bound for the equilibrium time ... 

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... 

It is important to note that the use of a cascade of columns of decreasing size, such as in Fig. 13.41, does not affect the consumption of chemicals for reflux, because this depends on the interstage flow required at the feed point. The cascade of columns of decreasing size does, however, reduce the total volume and the holdup of desired isotope. If the cascade of columns were not used for the N separation example, with its low feed concentration and separation factor close to unity, the holdup would be so great that product concentration would not reach... 

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. 

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