Cascade close-separation, ideal

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

The fact that the total internal flow rate in a close-separation, ideal cascade is given by Eq. (12.142) may be derived without solving explicitly for the individual internal flow rates by the following development, due originally to P. A. M. Dirac. This procedure is valuable in showing the fundamental character of the separation potential and the separative capacity, and provides a point of departure for the treatment of multicomponent isotope separation. 

We consider a close-separation, ideal cascade Wiose external streams have molar flow rates Xif (positive if a product, negative if a feed), and compositions X/t expressed as mole fraction. Let us look for a function of composition x ), to be called the separation potential, with the property that the sum over all external streams, to be called the separative capacity D,... 

In a close-separation, ideal cascade 6i =, so that the total flow leaving the rth stage is... 

The total inventory of the enriching section Ip then is just h times the total flow rate in the enriching section for a close-separation, ideal cascade. 

To make use of Eq. (12.209) we need expressions for the inventory of both components /, the inventory of desired component Ix and the inventory of separative work 70 in a close-separation, ideal cascade. To derive these expressions we shall assume that the stage inventory is proportional to the stage feed rate, as stated by (12.198), and that the average stage composition is that of the stage feed z,-. 

Heads flow rate. The flow rate M of stage heads of composition y in the enriching section of a close-separation ideal cascade producing product at rate P and composition yp is... 

In some isotope separation plants, notably those using distillation or exchange processes, it is more economic to use a constant interstage flow rate over a considerable composition interval rather than a flow rate that decreases steadily from the feed point to the product ends, as is characteristic of an ideal cascade. Cohen [C3] has called such cascades squared-off cascades and has derived equations for their separation performance. This section summarizes the derivation for a close-separation, squared-off cascade. 

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... 

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. 

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... 

Np and Ny/ are the minimum number of enrichment and stripping stages, respectively. In isotope separations a is often very close to one In a can then be replaced by (a — 1). In practice some product flow is desired, the fraction withdrawn at the enrichment stage being known as "the cut" PIF. The number of stages required to produce the composition Xp then increases. The most economic, and thus also the most common, type of cascade for processes with near unity a is the so-called ideal cascade. In this there is no mixing of streams of unequal conc trations, thus j in Fig. 2.8. Alfliougfa the number of... 

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