Minimum reflux cascade

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 two limiting cascades we have discussed are of little practical value. The minimum stage cascade produces zero amount of the maximum concentration material the minimum reflux cascade produces the maximum amount of material with no enrichment. Both of these cascades... 

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. 

When a lies close to 1 the minimum reflux ratio is large, but since Xj varies with stage number so does [n(i+1)/P]MiN- At the feed point in a 235U plant enriching to 90% 235U, (nf/P)MiN is 29,100, but at the product end of the cascade it approaches zero. 

A practical isotope separation plant can operate at neither minimum reflux (where the separation is zero, but the rate of production is high), nor at minimum number of stages (where the rate of production is zero, but the separation is high). A compromise is required. Since optimum reflux varies with stage number it is customary to employ tapered cascades for isotope separation. This results in marked savings in material hold-up, and in plant size and investment. 

Chempak . 427, 429. 500, 616, 645, 885 Chemstation, 180 Chien s minimum reflux, 103, 104 Chien s minimum stages, 105 CMR (see Cascade Mini-Ring) Colburn ... 

The minimum reflux ratio to establish a positive concentration gradient in the cascade follows from Equation 17... 

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. 

A. Acrivos and N. R. Amundson On the Steady State Fractionation of Multicomponent and Complex Mixtures in an Ideal Cascade Part 2—The Calculational of the Minimum Reflux Ratio, Chem. Eng. Sci., 4(2) 68 (1955). 

These equations all show that the minimum reflux ratio increases as the composition departs more from product or tails composition. In isotope separation cascades in which a is close to unity, the minimum reflux ratio is enormous. For example, at the feed point of a plant... 

Yet, as the product end of this cascade is approached, the minimum reflux ratio approaches zero. 

As in the fractionation of binary mixtures, ideal plates are assumed in the design of cascades, and the number of stages is subsequently corrected for plate efficiencies. The two limiting conditions of total reflux and minimum reflux are also encountered. 

The reader should be aware that the minimum reflux scenarios presented here are just one of three possible ways the minimum reflux limit can be obtained in distributed feed columns. The designs shown thus far all depicted minimum reflux when the vertex of the internal CS adjacent to the topmost rectifying section lies exactly on its profile, that is, a pinch occurs on the topmost rectifying CS. It is perfectly valid for the minimum reflux condition to be determined by the bottommost stripping profile, or indeed where the TTs of the internal CSs do not overlap one another. The latter case is shown in Figure 6.12 where the column reflux has been reduced and TTs cascade around one another, thereby limiting any further column reflux reduction. The general requirement for minimum reflux is however the same as for simple columns any reflux value below the minimum reflux value will lead to a discontinuous path of profiles, and minimum reflux is therefore the last reflux where a continuous path is still maintained. 

The number of stages in an ideal cascade may be evaluated by a procedure similar to that used in deriving Eq. (12.72) for the minimum number of stages at total reflux. ITie result is... 

Thus the number of stages required for a given separation in an ideal cascade is just twice the minimum number needed at total reflux minus 1. 

Here y-p is the isotopic composition at the top stage, P is the product withdrawal rate, and, as before, X,+i is the tails withdrawal rate at the (i + l)th stage. The ratio Xi+JP) is the reflux, which for a simple countercurrent cascade is independent of stage number (i.e., such a cascade is squared off). Note that /, and X4+1 approach each other as Xi+JP approaches infinity, which happens at total reflux. In that case, the number of stages required to carry out a given overall separation is a minimum and is given by the Fenske equation (Fenske 1932) ... 

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. 

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