Ind. Eng. Chem. Fundam. 1980, 19, 323-324
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On the Value Function and Separative Work in Isotope Separation In using the value function in calculations based on elements having large elementary separation factors, it is worth considering that the elements have optimum separative powers, dependent on the operating conditions. These optimal conditions are not necessarily suitable for operating in an ideal cascade of any conformation. An optimal cascade in which an optimum compromise is made between the separating power of the element, on the one hand, and mixing, on the other, is suggested.
Introduction In the development of the theory of isotope separation (Cohen, 1951) a need arose for a measure of separative work in order to compare the performance of different separating elements and different cascades of elements. Separative work was defined in conjunction with a definition of “value”. Value, like thermodynamic quantities (enthalpy, entropy) was defined as a product of an extensive quantity (mass) and an intensive quantity called the value function. For example, suppose we have an element separating L moles of material with a concentration of desired isotope x into OL moles with a concentration x’and (1- O)L moles with a concentration r”; the separative work, or the increase of value, dU, is given by d U = L[O.V(x’) + (1 - O)V(x”)- V ( X ) ] (1) where V(x)is the value function for a concentration x . 0 is the “cut” of the element. V is clearly a function of x only, so that the separative work of two different elements depends only on the flow, cut, and concentration changes effected by the two elements. The value function derived under this constraint has the form
where R = x / ( l - x ) , and the subscript 0 refers to material of zero value. If the point of zero value is arbitrarily placed at x = 0.5, the value function is simply V ( x ) = (2x - 1) In ( x / ( l - x ) ) (3) On the basis of this definition of separative work, ideal cascades were defined. In these cascades no re-mixing of separated streams takes place, so that the total separative power of the cascade is the sum of the separative powers of the individual elements in the cascade, i.e. d U = n6U (4) where d U is the total separative power of the cascade, 6U is the separative power of each element, and n is the number of elements making up the cascade. T h e Value Function The forms of eq 2 and 3 were derived for small overall separation factors but the equations are applicable in the case of a large separation factor. They do, however, have an apparent disadvantage. The separating power, 6U, is dependent to some extent on the concentration at which the element operates. In other words, if the number of elements required for a job is to be calculated from n = dUfGU, the value of n will depend to some extent on the value x at which 6U was measured. This is particularly noticeable in the case of elements with large overall separation factor and a cut not in the region of one-half, i.e., asymmetrically operating units.
Because of this, Bulang (Bulang et al., 1960) derived a form for the separating power of a centrifuge which is independent of the concentrations but depends instead on the element separation factors at the heads and tails compositions, LY and p respectively.
(5) This separating power postulates a different value function which, again, depends on the heads and tails separating factors. This means that the value function is defined in terms of a particular element so that two different cascades dividing equal quantities of feed into equal quantities of product and waste at equal concentrations would not do the same amount of separating work if they were not constituted of identical elements. It appears then that if the separating power of the separating element is to be independent of concentration, the value function cannot be dependent on concentration only. On the other hand, if the value function depends on concentration only, the separating power depends on the concentration. This has been discussed in a number of papers (Bulang et al., 1960; Ouwerkerk and Los, 1964; Apelblat and Lehrer, 1968; Olander, 1976; Kanagawa, 1977) and a method (Yamamoto et al., 1977) of resolving this dilemma has been put forward, requiring more than one value function. The question arises to what extent the dependence of the separating power on concentration is significant. Numerical Test It is characteristic of an ideal cascade that it is widest at the feed stage and it tapers rather rapidly toward the waste and product stages. This is true for both symmetric and asymmetric cascades. Most of the separating elements are in the region of the feed stage. If the separating power of the element is measured at the cascade feed concentration it is apparent that the error introduced in calculation of the number of elements using eq 4 will be due to a small number of elements operating at concentrations far from the cascade feed concentration and will therefore be small. In order to check this, the cascade described by Wolf (Wolf et al., 1976) was examined. This cascade is based on units each having a large overall separation factor, 5, and marked asymmetry, viz., a heads separation factor of 1.9 and a tails separation factor of 2.63. The cascade operates between a waste concentration of 0.001 and a product concentration of 0.9, with 1000 units of feed at a concentration of 0.002. Using the value function of eq 3, the separating power of an elementary unit with unit feed was calculated at three different feed concentrations. In Table I the cut was taken from the cut at the corresponding concentration in the cascade. The cut changes along the cascade in order to preserve the heads and tails separation factors, just as in an ordinary, symmetric ideal cascade. At the highest feed concentration in the table, the separating power is 8%
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Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980
Table I. Separating Power as a Function of Concentration of a Unit with Unit Feed Quality, Heads Separation Factor 1.9, and Tails Separation Factor 2.63 feed concn
product concn
waste concn
cut
separating powera
0.0026 0.3433 0.8254
0.005 0.5 0.9
0.00100 0.1667 0.6429
0.4 0.53 0.71
0.3117 0.30118 0.28557
a Separating power according t o Bulang et al., 0.27938 unit.
Table 11. Separating Power of Asymmetric Cascadea flow
quantity
concn
value function (eq 3)
product product waste waste waste feed
0.93 1.17 214.96 300.12 482.81 1000
0.92261 0.89627 0.00006 0.00008 0.00011 0.002
2.09475 1.70907 9.71994 9.43189 9.11291 6.18776
a Separative work, 3136.1 units; average product concentration, 0.9079 ; average waste concentration, 0.00009; “net” separative work, 3110.53 units; separative work per unit = 0.3117; cascade separative work + unit separative work = 10061 units; “net” separative work + unit separative work = 9979 units; total cascade flow = total units = 10094 units.
lower than at the lowest, which is close to the cascade feed concentration. The separating power, using eq 5 is lower than that according to the value function near the cascade feed concentration by 10%. Still using the original value function, the gross separating power of the cascade was calculated taking into consideration two product flows and three waste flows. This is given in Table 11. It turns out that the total separating power of the cascade is 10060 times the separating power of an elementary unit measured at the cascade feed concentration. The total cascade flow, which is the total number of elements in the cascade, is 10094 units. The difference between this number and the number dU/6U is 34 units, representing a 0.34% error. This error is insignificant if one considers that the inaccuracy in separating power as a function of inaccuracy in flow measurement is likely to be as large. Thus in this quite extreme case the value function may be used as originally derived. It appears that in any ideal cascade, symmetric or asymmetric, the separating power is sufficiently insensitive to concentration to allow the use of this value function. Separating Power and Mixing Returning to Table 11, the “net” separative work given there is the separative work that is required to obtain one
product stream and one waste stream in which the concentrations are the average product and waste concentrations of the cascade. It can be seen that the loss in separative work by mixing the product streams to make one product stream corresponds to 82 elements or 0.8%. Clearly, this loss is too small to be decisive in the choice between operating in a symmetric or an asymmetric ideal cascade. There is thus a trade-off between obtaining the maximum separating power from the elementary unit on the one hand and losses of separating power by mixing on the other. By limiting ourselves in cascade design to no-mixing cascades, we are led to asymmetric (Olander, 1976) cascades or variable separation factor cascades. The constraint of no-mixing is not necessarily an assurance of optimum use of the units. A situation can be conceived where the elements operate near the optimum conditions for maximum separative power, while some part of the separative work done is lost in mixing in the connecting piping. If the gain in separating power of the element is larger than the mixing losses it causes in the cascade, there is a net gain in separative work. We thus arrive at a scheme for an optimum cascade which is applicable for elements which are themselves small cascades like distillation columns, thermal diffusion columns, and centrifuges. The calculation of such cascades is not straightforward, though. It is a nonlinear optimization problem where the separating power of the elements in each stage is increased causing a concomitant loss of separative work by mixing. The optimization is over the cascade as a whole and cannot be done stage-by-stage. Acknowledgment I wish to thank Professor F. Klein of the Weizmann Institute for his criticism of the manuscript. Literature Cited Apelblat, A,, Ilamed-Lehrer, Y., J . Nucl. Energy, 22, 1 (1968). Bulang, W., Both,W., Jordan, I., Kolbe, W., Nam, E., Welge. K. H., 2.Phys. Chem. (FrankbH am Maln), 24, 249 (1960). Cohen, K., “The Theory of Isotope Separatlon as Applied to the Large Scale Production of UZ3’”, Chapter 1, McGraw-Hill, New York, 1951. Kanagawa, A., Yamamoto, I., Mizuno, Y., J . Nucl. Scl. Techno/.. 14, 282 (1977). Olander, D., Nucl. Sci. f n g . , 80, 421 (1976). Olander, D., Nucl. Techno/., 29, 108 (1976). Ouwerkerk, C., Los, J., Roc. U.N.Conf. Peaceful Uses Atomic Energy, 12, 367 (1964). Wolf, D., Borowitz, J. L., Qabor, A,, Shraga, Y., Ind. fng. Chem. Fundam., 15. .-, 15 . -(1976). ~
Yamamoto, I., Kanagawa, A.. J. Nucl. Sci. Techno/., 14, 565 (1977).
Israel Atomic Energy Commission Tel-Aviv, Israel
J. L. Borowitz
Received f o r review January 1, 1979 Accepted January 14, 1980
