In some stage processes, the heads and tails streams are separated in such a way that all portions of each stream have uniform composition. This occurs, for example, in a well-mixed electrolytic cell operated with steady flow of feed water and steady withdrawal of partially electrolyzed water. In other stage processes, the heads or tails stream may be withdrawn in such a way that the other stream changes progressively in composition during the separation process. This occurs, for example, when water flows through an electrolytic cell without mixing, and becomes progressively richer in deuterium, or when water is electrolyzed batchwise and becomes richer in deuterium as time goes on. These are examples of differential stage separation, in which successive small portions of one stream are removed from a second without mixing the second stream or giving the first stream further opportunity to exchange material with the second.

Two types of differential stage separation are illustrated in Fig. 12.10. In type A the stream being removed in small portions is depleted in the desired component, while the remaining stream becomes progressively enriched in this component; the concentration of

deuterium in batch electrolysis of water is an example of this type of differential stage separation.

In type В the stream being removed in small portions is enriched in the desired component, while the remaining stream becomes progressively depleted in this component. The flow of a mixture of 235 UF6 and 238 UF6 along the barrier of a gaseous diffusion stage is an example of this type of process. The small portions of gas that pass through the barrier are enriched in the desired component, U235 F6, and the remaining gas flowing along the upstream side of the barrier becomes progressively depleted in 235 UF6.

Equations relating the flow rates and compositions of feed and product streams in differential separation processes, first derived by Lord Rayleigh [Rl] for batch distillation, are often called the Rayleigh distillation equation. We shall derive some of these relationships for type В differential stage separation, using the nomenclature shown in Fig. 12.11.

At a point in the stage where a small amount of heads stream having flow rate dM’ and composition у is separated, the flow rate of the remaining depleted stream is changed by amount dN’ and its composition is changed by dx’. The material balance equation on total flow is

dM’=—dN’ (12.22)

and the material balance equation on flow of desired component is

у dM’ = —dix’N*) (12.23)

The result of elimination dM’ is

-y’dN’ = — d(x’N’)

dN’ dx’

N’ y’ — x’

Heads

The result of integrating this equation from the feed end of the stage at which the flow rate is Z and composition z to the tails end where the flow rate is N and the composition x is

(12.26)

This is the general form of the Rayleigh equation. When the relationship between у and x is known, the equation may be integrated graphically or numerically.

For a two-component mixture, the relationship between y’ and x may be expressed in terms of a local separation factor a’, defined as

,_//(! -/) “ *70 -*’)

in analogous fashion to the stage separation factor defined by (12.15). The result of using this equation to eliminate у from (12.26) is

(12.28)

A relation between the stage separation factor a and the local separation factor a may be obtained from Eq. (12.31) by using (12.12) to replace z by у and (12.15) to eliminate y:

When a’ — 1 < 1, as in separating uranium isotopes by gaseous diffusion, this equation reduces to

In this form it can be seen that a is greater than a’, and becomes much greater as 8 approaches unity. Thus, differential stage separation may be used to enhance the difference in composition attainable in simple stage separation.

For type A differential stage separation, a similar derivation leads to

(12.34)

Because of Eq. (12.13) defining r and (12.16) defining /3,

_ 1

ol/(a’-l)

The equation corresponding to (12.33), applicable when a — 1 < 1, is

(a’-l)lnfl