Research Article 
Corresponding author: Evgeny V. Semenov ( evsmv@bk.ru ) Academic editor: Georgy Tikhomirov
© 2023 Evgeny V. Semenov, Vladimir V. Kharitonov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Semenov EV, Kharitonov VV (2023) Calculation of the cost of enriched uranium products in multistream cascades of enrichment process. Nuclear Energy and Technology 9(1): 1925. https://doi.org/10.3897/nucet.9.100752

Modern uranium enrichment facilities can simultaneously use several raw materials as feed, including natural uranium, regenerated uranium obtained as a result of SNF reprocessing, or depleted uranium (all in the form of uranium hexafluoride). As the output of the separating cascade, several types of enriched uranium product with different levels of enrichment can be fabricated simultaneously. The paper proposes a methodology, absent in literature, for calculating the cost of each enriched uranium product in multistream separating cascades. The proposed methodology uses standard definitions of the isotopic value of feed and product stream and the PeierlsDirac separation potential. Numerical calculations of the cost of enriched uranium products for three production problems are provided as examples of the methodology effectiveness: 1) involvement of depleted uranium hexafluoride (DUHF) in fabrication of enriched uranium product; 2) simultaneous fabrication of two enriched products; 3) use of depleted uranium to reduce the cost of the product with a higher enrichment level out of two (as applied, e.g., to advanced tolerant fuel). It has been shown that partial additions of DUHF as feed for a multiproduct separating cascade make it possible to reduce the cost of a product with a higher level of enrichment; with the current market prices for natural uranium and separative work, there is a range of tails assays in which it is more profitable to enrich DUHF rather than natural uranium.
Enriched uranium product cost, multistream enrichment process, separative work, effectiveness of depleted uranium involvement in enrichment process
Modern facilities for separation of uranium isotopes are capable to use simultaneously more than one raw material as the separating cascade feed, including not only natural uranium but also regenerated uranium obtained as the result of SNF processing, or depleted (waste) uranium (all in the form of uranium hexafluoride). The separating cascade output may simultaneously include several types of enriched uranium product with different enrichment levels. While the cash value of raw materials is normally known (from market data or contract conditions), no calculation procedures to determine the cost of each of the enriched products could be found in available publications. Some publications deal with the topical problem of optimizing multistream cascades to clean regenerated uranium of ^{232}U, ^{234}U and ^{236}U isotopes which accumulate in the process of recirculating repeatedly regenerated uranium when using, e.g., REMIX fuel (
The purpose of the study is therefore to propose a methodology for calculating the cost of each enriched uranium product in multistream separating cascades.
Fig.
Flow diagram of separating cascades: А. Traditional cascade with one feed stream, F (c), of natural uranium and one stream of enriched uranium product, P (x), with a waste stream of D (у); В. Cascade with two feed streams and two streams of enriched uranium; С. A part of cascade В with the same two feed streams as in Fig. В but with one cascade outlet (intermediate) stream of product, Р_{х1}=Р_{1}(х_{1})+Р_{2}(х_{1})+ΔP, with enrichment х_{1}.
З = FЦ_{F} + RЦ_{R} + DЦ_{D}, (1)
including the three above types of costs. We shall note that the cost of separative work, as shown in (
The generally accepted definition of separative work is expressed as the difference between the isotopic value of products (enriched uranium and tails) and the raw material (natural uranium) in the form shown in (
R = P (x)V (x) + D (y)V (y) – F (c)V (c), (2)
where V (z) = (1 – 2z)ln[(1 – z)/z] is the PeierlsDirac separation potential (z = x, y or с).
Isotopic value is the product of the stream weight by the stream separation potential (e.g., the value of the enriched product is Р (х)V (х)). The relationship among streams F, P and D follows from the balance of the masses as coefficients of the natural uranium consumption per the product unit, (F/P), and the mass formation per the product unit, (D/P)
F/P= (x – y)/(c – y); D/P= F/P– 1 = (x – c)/(c – y). (3)
Substituting (3) in (2) leads to the classical definition of specific separative work (also referred to as separative work standard)
R/P= V (x) + V (y)(x – c)/(c – y) – V (c)(x – y)/(c – y). (4)
Substituting expressions (2) through (4) in formula (1) leads to the traditional definition of costs, З, for the enrichment of natural uranium and the enriched uranium product cost, C_{х}($/kg),:
С_{x}= З/P = Ц_{F} (x – y)/(c – y) + Ц_{R}[V (x) + V (y)(x – c)/ (c – y) – V (c)(x – y)/(c – y)] + Ц_{D} (x – c)/(c – y). (5)
The only parameter, using which the product cost can be handled with the given product enrichment, x, and determined prices for the raw material (natural uranium with a concentration of с =0.711%), separative work and tails recycling, is tails assay у (concentration of ^{235}U in uranium tails). As shown in (
Fig.
By generalizing the approach in (
〈x〉 = (x_{1}P_{1} + x_{2}P_{2})/P; 〈c〉 = (c_{1}F_{1} + c_{2}F_{2})/F; P = P_{1} + P_{2}; F = F_{1} + F_{2}; (6)
〈V (x)〉 = [V (x_{1})P_{1} + V (x_{2})P_{2}]/P; 〈V (c)〉 = [V (c_{1})F_{1} + V (c_{2})F_{2}]/F; (7)
Ц_{F}〉 = [F_{1}Ц_{F1} + F_{2}Ц_{F2}]/F = fЦ_{F1} + (1 – f)Ц_{F2}. (8)
Here, Р and F are the cumulative masses of the enriched product and feed streams respectively; and f = F_{1}/F is the share of the cascade feed with a smaller concentration of ^{235}U. Expressions (6) and (7) follow unambiguously from the classical definition of separative work with regard for the isotopic value of all feed and removal streams looking as follows for flow diagram В:
R_{B} = P_{1}V (x_{1}) + P_{2}V (x_{2}) + D (y)V (y) – F_{1}V (c_{1}) – F_{2}V (c_{2}). (9)
For a multiproduct cascade, as a result, we obtain analogs of expressions (4) and (5) for specific separative work and unit costs as follows
R_{B}/P = 〈V (x)〉 + 〈V (y)〉(〈x〉 – 〈c〉)/(〈c〉 – y) – 〈V (c)〉 (〈x〉 – y)/(〈c〉 – y) ≡ V_{B}; (10)
З_{B}/P = 〈Ц_{F}〉(〈x〉 – y)/(〈c〉 – y) + Ц_{R}V_{B} + Ц_{D} (〈x〉 – 〈c〉)/(〈c〉 – y). (11)
Value V_{B} in expression (10) is the specific separative work or the separative work rate for a multiproduct separating cascade. It follows from expression (10) and (11) that, with the given feed and product removal parameters, there is an optimum tails assay, у_{0}, with which the enriched product fabrication costs are as small as possible, while, as in the previous case (flow diagram А), value у_{0} depends only on the price ratio. In particular, if the cascade feed consists of natural uranium (с_{2} = с = 0.711%) and depleted uranium hexafluoride (DUHF) streams with a zero value (Ц_{F}_{1} = 0), which have equal masses, then we get for the above market prices that (Ц_{F}_{2} ≡ Ц_{F}, F_{2}/F = F_{1}/F = 1/2)
〈Ц_{F}〉= Ц_{F}/2; (〈Ц_{F}〉+ Ц_{D})/Ц_{R} = 1.12 and y_{0} = 0.216%.
Expression (11), with the given enriched product weight, Р, defines the cumulative separation costs for purchasing two types of raw materials for the cascade feed (the first term in the second member) and for separative work with two enriched uranium product removals (the second term) taking into account the tails recycling costs (the final term). The costs for each raw material type in formula (11) have been determined, while the isotope separative work costs for each enriched product, as well as the raw material value contribution to the cost of each enriched product have not been determined, this making it impossible to estimate the cost of each enriched product. We shall use flow diagram C to solve this problem.
The theory of separating cascades (
F/(P + ΔP) = (x_{1} – y)/(〈c〉 – y) or ΔP/P = β_{2} (x_{2} – x_{1})/(x_{1} – y). (12)
In the above expression, β_{2}=Р_{2}/Р demotes a mass fraction of a product with a higher enrichment level in the separating cascade’s product portfolio and takes into account that, according to (10),
F/P = (〈x〉 – y)/(〈c〉 – y) и 〈x〉 – x_{1} = β_{2} (x_{2} – x_{1}).
It follows from (12) that ΔР = 0 with х_{1} = х_{2} and β_{2} = 0. If β_{2} = 1, then, simultaneously, х_{1}=х_{2}. We obtain the separative work for flow diagram C, by analogy with standardly derived formulas (4) and (10), as follows
R_{C}/(P + ΔP) = V (x_{1}) + V (y)(x_{1} – 〈c〉)/ (〈c〉 – y) – 〈V (c)〉(x_{1} – y)/(〈c〉 – y) ≡ V_{C}, (13)
where RС is the separative work upstream of the enriched uranium first removal stage in flow diagram С; and V_{C} is specific separative work (or the separative work rate) in flow diagram С.
Difference ΔR = R_{B} – R_{С} between separative work (10) for flow diagram В and separative work (13) for flow diagram С leads to the amount of separative work per the increase of the enrichment of uranium mass P_{2} + ΔP from х_{1} to х_{2} to obtain the second product of mass Р_{2}. It remains only to distribute, among each commercial product, separative work, R_{C}, as well as the feedstocks in flow diagram B defined by the first term in formula (11), and the tails recycling costs defined by the final term in formula (11). We shall denote the total of the feedstock and tails recycling costs as:
З_{BFD} = P[〈Ц_{F}〉(〈x〉 – y)/(〈c〉 – y) + Ц_{D} (〈x〉 – 〈c〉)/(〈c〉 – y)]. (14)
As part of the traditional approach to the separative work calculation under consideration, it appears logical to distribute values З_{BDF} and R_{C} in proportion to the mass fraction of each enriched product. As the result, the cumulative costs for each enriched product are determined as:
З_{1} = β_{1} (З_{BDF} + R_{C}Ц_{R}); З_{2} = β_{2} (З_{BDF} + R_{C}Ц_{R}) + ΔR_{C}Ц_{R}, (15)
where b_{1} = P_{1}/P and b_{2} = 1 – b_{1} = P_{2}/P are relative mass fractions of removed enriched products. Expression (15) satisfies the condition З_{1} + З_{2} = З_{В}. As a result, the cost of each enriched product is determined
C_{x1} = З_{1}/P_{1} = (З_{BDF} + R_{C}Ц_{R})/P = = 〈Ц_{F}〉(〈x〉 – y)/(〈c〉 – y) + Ц_{D} (〈x〉 – 〈c〉) /(〈c〉 – y) + V_{C}Ц_{R} (〈x〉 – y)/(x_{1} – y); (16)
C_{x2} = З_{2}/P_{2} = C_{x1} + Ц_{R}ΔR/P_{2}. (17)
As can be seen from (17), the cost of a product with a large enrichment exceeds the cost of a product with a smaller enrichment by
Ц_{R}ΔR/P_{2} = Ц_{R}[PV_{B} – V_{C} (P + ΔP)]/P_{2} = = Ц_{R}[V (x_{2}) – V (x_{1})(x_{2} – y)/(x_{1} – y) + V (y)(x_{2} – x_{1})/(x_{1} – y)]. (18)
Hence it follows that ΔR = 0 with х_{1} = х_{2}. Therefore, the cost of each enriched product has been determined. The obtained results can be used to solve a number of manufacturing tasks. Presented below as examples are numerical calculations of the enriched product cost for the three following problems: involvement of depleted uranium hexafluoride (DUHF) in fabrication of enriched uranium product, 2) simultaneous fabrication of two enriched products, and 3) involvement of DUHF for reducing the value of a product with a higher enrichment level out of two (as applied to advanced tolerant fuel).
By now, there is some 2 million tons (Mt) of uranium hexafluoride with a ^{235}U concentration in a range of 0.1 to 0.4 wt.% (in terms of uranium metal) accumulated worldwide (
The concentration of ^{235}U in DUHF is smaller than in natural uranium, so more separative work, than in the event of only natural uranium enrichment, will be required for a greater output of the given amount of enriched uranium. But the cost of DUHF (Ц_{F1}) can be much below the natural uranium price (Ц_{F2}), so one can hope that the economic effect is positive.
It follows from Fig.
Cost of 4.95% enriched uranium product obtained from DUHF with a concentration of у_{1}=0.3% (a) and 0.2% (b), as a function of the secondary tails assay (у) and the DUHF cost (С_{D}= 0 to 10 $/kgU), with market quotes for natural uranium hexafluoride (C_{F}=90 $/kgU) and for separative work (C_{R}=45 $/SWU). The NU dashed line shows the cost of EUP from natural uranium.
In the limiting case of the DUHF zero price, it is more profitable to enrich DUHF rather than natural uranium practically across the tails assay range (up to 0.3% with the DUHF enrichment level of 0.3%) (Fig.
With the given installed capacity of the separating cascade, the generated EUP mass (per separative work unit, P/R) increases with the ^{235}U concentration growth in the tails (Fig.
Ratio, Р/R, of the enriched product mass to the separative work costs (in kgU/SWU) with enrichment of up to х=4.95% for natural uranium (f=0) or depleted uranium hexafluoride (DUHF, f=1), the content of uranium235 in DUHF being 0.3% and 0.2% depending on the separative cascade tails assay (у = 0.08 – 0.31%).
Let us estimate the dependence of the cost of each of the products with enrichments х_{1} = 2.5% and х_{2} = 4.95% (e.g., for RBMK and VVER reactors) on the mass fraction, β_{1}=Р_{1}/Р, of a low enriched product and the tails assay, у = 0.05 – 0.35%. The fraction of the low enriched product is variable in a range of β_{1} = 0 (flow diagram A, only EUP with an enrichment of х_{2} = 4.95% is fabricated) to β_{1}=1 (flow diagram А, only EUP with an enrichment of х_{1} = 2.5% is produced). In the latter case, when the cascade product consists of only low enriched uranium (х_{1} = 2.5%), its value is determined by expression (5) with х changed for х_{1}. It follows from Fig.
It is possible to check in this section to which extent the cost of a high enriched product can be reduced (e.g., with х_{2} = 7% as for advanced tolerant fuel with a 24month fuel cycle (StateoftheArt Report,
Cost price of enriched uranium products in a twoproduct separating cascade with х_{1} = 4.95% and х_{2} = 7% depending on the fraction of a high enriched product (β_{2} = Р_{2}/Р =0 – 1) and the fraction of DUHF with a concentration of c_{1} = 0.3% in the cascade feed (f = 0 – 1). Initial prices: Ц_{F1} = Ц_{D} =0, Ц_{F2} = 90 $/kgU, Ц_{R} = 45 $/SWU. Tails assay: у = 0.15%
Cost $/kgU  Natural uranium feed f = 0, <c> = 0.711%  DUHF feed f = 1, <c> = 0.3 %  

β_{2}=0  β_{2}=0.2  β_{2}=0.8  β_{2}=1  β_{2}=0  β_{2}=0.2  β_{2}=0.8  β_{2}=1  
С _{х1}(х_{1}=4,95%)  1219  1323  1635  –  850  923  1141  – 
С _{х2}(х_{2}=7%)  –  1377  1689  1793  –  978  1195  1267 
Cost of enriched uranium products for a twoproduct separating stage with х_{1}=4.95% and х_{2}=7% depending on the fraction of DUHF with a concentration of c_{1}=0.3% in the cascade feed (f = 0 – 1) and the fraction of high enriched product (β_{2} = Р_{2}/Р), equal to 0.2 (а) and 0.8 (b), with a tails assay of у = 0.15%. Thicker lines match the limiting values of f = 0 and f = 100%. Initial prices: ЦF_{1} = ЦD = 0, ЦF_{2} = 90 $/kgU, ЦR = 45 $/SWU.
As it follows from the table, the cost of an enriched uranium product in a oneproduct cascade with only natural uranium feed (f = 0) amounts to 1219 $/kgU (х_{1} = 4.95%) and 1793 $/kgU (х_{2} = 7%) with the same tails assay, у=0.15%. The cost of the same products with the oneproduct cascade feed of only DUHF (f = 1) with the ^{235}U isotope concentration of с_{1} = 0.3% and of zero value decreases respectively to 850 and 1267 $/kgU (by about a factor of 1.42). Simultaneous manufacturing of products with different enrichments in a twoproduct separating cascade with only natural uranium feed leads to a reduction in the price for a product enriched to a higher level (from 1267 to 978 $/kgU) and an increase in the price of a product enriched to a lower level (from 850 to 923 $/kgU). And the price of a 4.95% enriched product remains much below 1219 $/kgU which corresponds to an only natural uranium cascade feed. The obtained results show that it is possible to reduce considerably the cost of enriched uranium products in a multiproduct separating cascade when using depleted uranium hexafluoride as the cascade feed.
The calculation results presented in Fig.
A decrease in the fraction of DUHF in the cascade feed leads to an increase in the cost of both products this increase being the greater the greater is the tails assay. An increase in the fraction of a product with a higher enrichment level (up to 80%) leads to a growth in the cost of both products (Fig.
Effects of the DUFH fraction (f) in the feed for a twoproduct separating cascade and the fraction (β_{2}) of a high enriched product (х_{2} = 7%) on the cumulative output of enriched uranium products (Р, tons) per 1 million separative work units (R, mln SWU) and on the feed demand (F) per product unit (Р) with х_{1} = 4.95% and a tails assay of у = 0.15%
f  β_{2}=0  β_{2}=0,2  β_{2}=0,8  β_{2}=1  

P/R t/mln SWU  F/P  P/R t/mln SWU  F/P  P/R t/mln SWU  F/P  P/R t/mln SWU  F/P  
0.0  100  7.6  90  8.3  70  10.5  65  11.2 
0.2  95  9.0  86  9.9  66  12.5  62  13.3 
0.5  84  12.5  76  13.6  59  17.1  55  18.3 
0.8  69  19.7  62  21.4  49  26.7  45  28.5 
1.0  53  31.0  48  33.7  38  41.9  35  44.7 
A novel methodology is presented for calculating the distribution of costs for each enriched product and, accordingly, the cost of each product in a multistream separating cascade. The methodology uses the standard definition of separative work and the Peierls and Dirac separation potential.
The results obtained based on the proposed methodology are presented for numerical calculations of the enriched uranium product cost for three fabrication problems: 1) involvement of depleted uranium hexafluoride (DUHF) in fabrication of enriched uranium product, 2) simultaneous fabrication of two enriched products, and 3) use of DUHF for reducing the cost of the product with a higher enrichment out of the two (as applied to advanced tolerant fuel).
It has been shown that manufacturing of two products (as compared with manufacturing of one product) makes a more expensive product cheaper and, vice versa, makes a cheaper product more expensive. Additions of DUHF as a feed for a multiproduct separating cascade make it possible to reduce the cost of a product with a higher level of enrichment and to increase to a certain extent the cost of a product with a lower level of enrichment. It has also been shown that the existing market prices for natural uranium (in the form of uranium hexafluoride) and separative work lead to a separating cascade tails assay range in which it is more profitable to enrich DUHF rather than natural uranium.