The cost of enriched uranium is based on the cost of U308, the conversion of U308 in UF6 and the enrichment costs. The enrichment costs are expressed in units of separative work which give a measure of the required isotopic separation effort."7’ The separation work is related to the quantities of material fed to and withdrawn from the process and their enrichment. An enrichment plant splits the fed uranium with enrichment e„ in two quantities, the product with enrichment eP and the tails (depleted U) with enrichment є,.

The following considerations are based on an ideal diffusion cascade. If we consider the balance of the mass of 235U and 238U in an enrichment plant we have

M„e„ = M„e„ + M, e, 235U balance,

238 (10.22) M„(l —e„) = Mp(l —eP) + M,(l-e,) 238U balance

where M, is the mass of natural uranium with enrichment e„, Mp is the mass of product uranium with enrichment eP, M, is the mass of tails with enrichment e„

out of (10.22) we obtain

M„ є„ — e, , M, eP — e„

TT =———— and TT =—————- •

Mp e„ ~ є, Mp e„ — Є,

The plant consists of a series of stages.

For each stage we can define

€г, feed stream enrichment,

ex, enrichment of the depleted stream leaving the stage, tv, enrichment of the enriched stream leaving the stage.

A stage separation factor is defined as

as a is very near to unity the quantity

ф = a — 1

is also often used. From (10.24) and (10.25) we have

fed + Ф)

Єу 1 + фех ‘

since ф < 1 it is possible to expand this equation in a series

Єу = Єх( + ф)( — фех + ф2ех2+ ■ ■ •)

and ignoring the terms in ф2 or higher powers

Єу — ex = фех(1 — єх). (10.28)

Having these preliminary definitions it is possible to look for a “value function” V(e) giving the value of a unit of uranium as a function of its enrichment є. This value function can then be used to define the increase in value effected by each stage and hence give a measure of the separative work.

The increase in value effected by a certain stage should be independent of the enrichment of that stage, because the work done by a stage is only dependent on the amount of material, but not on its enrichment. The value function we are looking for should satisfy this condition. Assuming the existence of this function, the increase in value effected by the stage is

A = MV(ex) + MV(ey) — 2MV(f:) (10.29)

and this A is defined to be the “separative work” done by the stage under consideration. We have here assumed that the mass 2M entering the stage is equally subdivided between enriched stream and depleted stream (it is possible to obtain the same definition of V(e) without this assumption).

2M, «

Expanding V(ez) and V(ey) in a Taylor’s series V(6,)= У(0 + (е, — е,)УЧО + к y 2 x> V"(ex) + — ■ ■ (10.30) V(0= V(e,) + (ez — е,)УЧе,) + (бг ~2Єх) V"(ex) + — ■ ■ (10.31) And from a stage 235U balance we have ez — ex = (ey — ex)l2. (10.32)

Substituting (10.30), (10.31) and (10.32) in (10.29) and neglecting the powers higher than two of the small difference (ey — є,) we have

A = M(ey — ex)2 V"(ex)l4. (10.33)

Substituting (єу — є») by expression (10.28) we obtain

A = ^[ex(l-ex)fV"(ex). (10.34)

A must be independent of enrichment. This means

^) = ІІЛТ^)Г (1035)

The solution of this differential equation is

V(ex)=C0+Ciex+(2ex — 1)In (737)- ПО-36)

If we set = 0 the two arbitrary constants Co and Ci we get a particularly convenient form of the “value function”,

Substituting eqn. (1037) in (1034) we have an expression for the “separative work" (or increase in value) effected by a stage, which satisfies the condition of being independent of enrichment. The total separative work done by the plant is the sum of the As corresponding to all individual stages.

Separative work = ^ Д for ideal plant.

All stages

For a real plant 2Д is greater than for an ideal plant, but the amount of separative work

All stages

delivered is the same because the plant does the same job.

Considering the value balance of the entire plant we have

Total separative work = MpV(ep) + M, V(et) — M„V(e„). (10.38)

As the value function is dimensionless, the separative work is expressed in kilograms of uranium. The number of units of separative work to obtain a unit mass of uranium with enrichment ep is

w = V(ep)’+W V(e,)~W V(en)- (l0-39)

The ratios Mp/M„ and M,/M„ can be obtained from (10.23) and the value function from (10.37), so that we have

Cp = Cf + WCS (10.41)

where Cf = unit cost of U in the form of UF6, Cs = unit cost of separative work, and from

(10.22)

Cp = 6p ~6> Cf + WCS. (10.42)

€n

The unit cost of separative work Cs is calculated in such a way as to cover all costs of the separation plant.

An example of the cost of enriched 235U as a function of enrichment is given in Fig. 10.4.<M) This cost is, of course, dependent upon the value of the parameters specified in the figure and is therefore subject to changements.

The price of other fissile materials like 233U or Pu can be derived comparing their neutronic worth with the one of 235U. If the bred fuel is reprocessed and burnt in the same reactor it is not necessary to put a price on it. If it is burnt in a different reactor its price can be defined as the one giving the same fuel-cycle costs to the two symbiotic reactors.

When the bred fuel is recycled in the same reactor it is also possible to define its price as the price of the amount of enriched uranium which would be required to substitute the unit weight of the fuel under consideration. (This value is sometimes called “indifference value” because it makes the fuel cycle cost indifferent to the decision of recycling or selling the bred fuel.)

The value of the bred fuel is dependent upon the percentage of undesirable isotopes.

Fig. 10.4. From GA-A10593.

For example the 235U discharged from HTRs contains considerable amounts of 236U which is a parasitic absorber, therefore the “indifference value” of this discharged fuel must be calculated considering the 236U concentration. Another example is given by the bred 233U which is accompanied by 232U and its daughters which are у emitters involving additional handling costs. Even without 232U the handling costs of 233U and of Pu are higher than for 235U because glove boxes are required for the manipulation of these isotopes.

10.13. Methods for fuel-cycle cost calculations

The easiest way to calculate fuel-cycle costs is the so-called “Discounted cash flow” method which consists of present-worthing the cash flow X and the energy E:"4’18’

F(t)X(t)

C =——————- (10.43)

2 F(t)E{t)

where

F(t) = (1 + /) (r ro>

is the present worth factor with the reference time t0 usually chosen as the reactor commercial start up.

The summations of eqn. (10.43) extend from the time of the first payments before the reactor start up to the first 10, 20 or 30 years of operation. The cash flow X must consider the expenditure for fissile material, enrichment, fertile material, shipping, fabrication and reprocessing: if bred fuel is being sold also these revenues must appear in the cash flow. At the end of the accounting period the value of the fuel present in the reactor must be assessed and credited to the cash flow.

This method has some disadvantages.

1. It gives only an average fuel-cycle cost and not a cost per reload interval or per segment.

2. The same interest rate must be used for the pre-irradiation, in-core, and post­irradiation time intervals.

3. It requires the evaluation of a final settlement at the end of the period.

These disadvantages are avoided by the “Discounted energy cost” method."4’ In this method each segment is divided into several accounting periods, usually corresponding to the reloading intervals. It is then possible to calculate the fuel-cycle cost of each segment s for a given reloading interval n. Let P,„ be the change in value of the segment s during the reloading interval n present-worthed to the midpoint (or to the beginning) of the interval.

The law according to which the fuel value is assumed to change with time during its residence in the core can only be defined in an arbitrary way because only the values before and after irradiation are known. It is possible to assume that the fuel value decreases linearly as function of the produced energy, or as function of burn-up. The Dragon KPD code"9’ assumes that the fabrication costs are completely assigned to the time interval at the beginning of which the segment is loaded, and that the reprocessing expenditures are assigned at the interval at the end of which the segment is discharged.

Let £sn be the energy produced by the segment s in the interval n present-worthed to the midpoint (or to the beginning) of the interval. Here again only the total energy (E„) produced by the core during the interval n is known, and the energy Es„ can be defined as the fraction of E„ corresponding to the volume fraction occupied by the segment s in the core.

En = 2 £„.

5

The fuel-cycle cost associated to the segment s in the interval n is

C„=!^. (10.44)

c,„

The fuel-cycle cost during the reloading interval n is

(10.45)

It is also possible to calculate C„ directly as

2 P

C„ = ——

En

without the need of defining the energies per segment £,„. This procedure is used by the KPD code."91

If the C, n have been calculated it is possible to define the average fuel cycle cost for segment s: 2 C, nEsnF„

2 E, nFn

where F„ is the present-worth factor for interval n.

The present-worthed levelized cost is given by

The differences between C, C„ and Cs are due to the fact that, especially during the running-in phase, the amount of fissile material consumed and the capital charges accumulated before producing energy can be different from one segment to another.

In all the above formulas we have not specified whether the energy E is an electrical or thermal energy.

If the product which is being sold by the plant is process heat E will be a thermal energy, while if the product is electricity E is an electrical energy. The thermal energy produced by the core in a given time interval can be easily calculated from the number of fissions which have taken place. The electrical energy is then calculated multiplying the thermal energy by the net plant efficiency.

10.14. Simplified calculations for the equilibrium cycle

With the above described method it is possible to obtain the fuel-cycle cost C„ for any reloading interval n. This cost will remain constant once equilibrium is reached and its value will give the cost of the equilibrium cycle.

For survey studies simplified methods are often used."8’ We can think that the fuel cycle cost C is composed of a running cost Cr (out-of-pocket cost without capital charges) and a cost C, due to interest on investment.

Considering a fuel segment

(10.50) where A = cost of the fuel charged in the reactor (including fabrication),

В = value of the discharged fuel (after deduction of shipping and reprocessing costs),

E = energy produced by the segment.

The calculation of the investment cost requires an evaluation of the value of the fuel present in the reactor.

A simplification which is often used consists in assuming that this value is<20)

A+B

2

so that

(10.52)

where Ta is an equivalent pre-irradiation time, Ть is an equivalent post-irradiation time, T is in-core residence time, і is interest rate.

While the running cost is independent of time but only depends on the fuel consump­tion, in eqn. (10.52) appears the residence time. A low load factor of the plant increases the residence time and hence the interest to be paid.