10.7. From the economic standpoint, the user of fuel needs to know how much uranium feed is required to produce a certain amount of enriched material and the cost of enrichment. To show how these quantities are determined, general relationships will first be derived between the amounts and assays of feed, enriched product, and tails. The following treatment is applicable to both gaseous-diffusion and gas-centrifuge processes al­though it has hitherto been employed primarily for the former.

10.8. If Fis the mass of uranium (regardless of its isotopic composition) in the feed material supplied to a separation cascade, P is the mass of product withdrawn, and W is the mass of the waste, a uranium mass balance (Fig. 10.1) requires that

F = P + W, (10.1)

assuming, as is generally true, that there is no appreciable loss of uranium during the operation. A similar balance can be applied to the uranium-235 only; thus,

F{xf) = P(xp) + W(xw),

PRODUCT

where Xf, xp, and xw are the assays (expressed as mass fractions of uranium — 235) in feed, product, and waste, respectively. By eliminating Wfrom these two equations, the result is

This equation gives the mass of uranium feed of assay xf required per unit mass of uranium product of assay xp, assuming a tails (waste) assay of xw.

10.9. The cost of enrichment is determined by the amount of work that has to be done to achieve the enrichment. A so-called value function has been developed on the basis of the theory of the gaseous-diffusion cascade. It is represented by V(x) and is defined for any material of assay x by

V(x) = (1 — 2x) In (10.4)

Because x is a fraction, the value function, which is characteristic of a given assay (uranium-235 content), is a fraction and has no units. It is used to determine the work required to yield a product of a desired assay from a given feed with a specified waste.

10.10 The effort expended in separating a mass F of feed of assay xf into a mass P of product of assay xp and waste of mass W and assay xw is expressed in terms of the number of separative work units (SWU) needed. This is given in terms of the respective value functions by

SWU = WV(xw) + PV(xp) — FV(xf). (10.5)

Since the value functions have no units, the SWU will have the same units as the masses W, P, and F. The general practice is to state the number of separative work units in terms of kilograms of uranium. Upon dividing equation (10.5) through by P, the result

SWT I W F

— = J V(xw) + V(xp) — — V(xf) (10.6)

gives the number of separative work units received per unit mass of product.

Example 10.1. The owner of a nuclear power plant requires 100,000 kg of 3.0 percent enriched uranium. Determine the amount of natural uranium feed and the number of separative work units needed, assuming a tails assay of (a) 0.200 percent and (b) 0.300 percent.

First, the necessary value functions are determined from equation (10.4); the results are as follows:

	X	V(x)
Product	0.03	3.27
Feed	0.0071	4.87
Tails	(a) 0.002	6.19
	(b) 0.003	5.77

{a) For xw = 0.002, equation (10.3) gives

F = 0.0300 — 0.0020 _ , P ” 0.0071 — 0.0020 “ ‘

and from equation (10.1)

W _ F P ~ P

Since F/P is 5.48, the natural uranium feed requirement is (5.48) (100,000) = 548,000 kg.

From equation (10.6),

SWU

—p — = (4.48)(6.19) + 3.27 — (5.48)(4.87) = 4.31.

Hence, the SWU requirement is 431,000 kg.

(b) For xw = 0.003, FtP = 6.57 and W/P = 5.57; hence, the natural uranium feed requirement is 657,000 kg.

The separative work is obtained from

SWU

—— = (5.57)(5.77) + 3.27 — (6.57)(4.87) = 3.42.

The SWU requirement is thus 342,000 kg.

10.11. The cost per separative work unit is determined from the en­richment plant operating costs and the cost of the capital invested in the plant. If Cs is the cost of a separative work unit, e. g., in dollars per kilogram, then

and

where Cp, Cf, and Cw are the costs per kilogram of product, feed, and waste, respectively. According to equation (10.7), the cost of the product is equal to the cost of the enrichment operation (separative work cost) plus the cost of the feed, less credit for the value of the waste (tails). In practice, the credit for the tails is usually neglected. The unit cost of the product then depends on the ratios SWU IP and F/P, which are determined by the assays of the product, feed, and tails, and on the costs per kilogram of the separative work and the feed.