Research Article 
Corresponding author: Evgeny V. Semenov ( evsmv@bk.ru ) Academic editor: Yury Kazansky
© 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) Analytical dependence of burnup on enrichment of prospective fuel and parameters of reactors fuel campaign. Nuclear Energy and Technology 9(4): 239244. https://doi.org/10.3897/nucet.9.116653

The paper is devoted to the definition of an analytical expression for estimating the burnup depth of nuclear fuel depending on its enrichment level, the periodicity of refueling, thermal stressthermal stress and the duration of the time period between refueling (reactor campaign) in a wide range of changes in key parameters for different types of thermal neutron reactors. The analytical expressions obtained in the work for the burnup depth are compared with numerous neutron physics calculations and experimental data from different authors for uranium fuel enrichment up to 9%. Calculations of the fuel share of the cost of electricity of nuclear power plants with PWR type reactors were performed and its sensitivity to changes in burnup depth and enrichment of fuel, the refueling periodicity, as well as to market prices for natural uranium, conversion, enrichment, fabrication of fuel assemblies and SNF handling were determined.
NPP, nuclear fuel burnup, enrichment, refueling periodicity, reactor campaign, fuel share in electricity cost
An important energy and economic characteristic of nuclear fuel is the socalled fuel burnup (or specific energy yield) that influences the NPP economic performance (
To identify analytical possibilities for selecting economically feasible parameters of the fuel cycle of NPPs with an extended reactor campaign, we consider three approaches to the evaluation of nuclear fuel burnup.
Firstly, the uranium fuel burnup, B, (MW·day/kgU) is connected with the reactor operating time, T, (days) with thermal power Q to the replacement of N fuel assemblies (FA) via a known expression (
B = (Q×T)/(N×M_{FA}) = q∙n×T, (1)
where М_{FA} is the mass of uranium in each FA (kgU); n = N_{CORE}/N is the refueling ratio; and N_{CORE} is the number of FA in the reactor core. Quantity q = Q/M_{CORE}, where M_{CORE} = N_{CORE}×M_{FA} is the mass of fuel in the reactor core, is referred to as specific thermal stressthermal stress of fuel (about 40 kW/kgU for UO_{2}), and relation NМ_{FA}/T = Q/B = P represents the reactor fuel demand (kg/day or kg/year depending on the dimensionality used for В). As it follows from Fig.
Number of replaceable FAs (N) as a function of reactor campaign (Т, days) and fuel burnup (В, MW∙day/kgU) with the installed reactor thermal power (Q = 3200 MW), mass of fuel in each FA (М_{FA} = 470 kgU), number of FAs in core (N_{CORE} = 163), and maximum theoretical ICUF (Т/(Т + ΔТ)), where ΔТ = 32 days is the time of the reactor outage for refueling and repair. Calculation based on formula (1). Vertical dashed lines are the boundaries of the actual reactor operating times to refueling in 12 and 24month cycles.
Secondly, back in the 1950s, the concept of an ideal fuel reloading regime was introduced, in which the reactor is fed with fresh fuel in microdoses with mixing throughout the entire core volume (
B (n) = В∞ n/(n + 1) (2)
Normally, n = 3−5, so the burnup is 75 to 83% of the ideal burnup.
By excluding n from formulas (1) and (2), we find the dependence of burnup on the fuel thermal stressthermal stress and the reactor campaign as the remainder in the following
B=B_{∞} − qT (3)
Thirdly, burnup can be expressed in terms of the mass of the nuclides burnt during the reactor campaign. Since the thermal energy generated for the reactor campaign, Q = ΔM_{f} (E_{f} /m_{f}), is directly proportional to the mass of the nuclides burnt, ΔM_{f}, which is nearly equal to the fission product mass, expression (1) for the fuel burnup can be reduced then as follows
B = (ΔM_{f} /M_{5})(M_{5}/M_{F})(E_{f} /m_{f}) = (E_{f} /m_{f}) (ΔM_{f} /M_{5})x. (4)
In the obtained expression (4), M_{F} = N∙M_{FA} is the mass of the fuel extracted during refueling; М_{5} is the mass of uranium235 in the fresh fuel loaded into the reactor instead of the spent fuel during reactor campaign; x = M_{5} /M_{F} is the fresh fuel enrichment; E_{f} /m_{f} = 970 MW∙day/kgU is the average caloric value of fissionable nuclides (uranium and plutonium) with an error of ± 1% (with the uranium and plutonium caloric values being in accordance with data in
Using enrichment in %, as generally accepted, expression (4) can be written as
B (MW×day/kgU) = 9.7x (%)∆M_{f} /M_{5} (5)
As can be seen, nuclear fuel burnup is directly proportional to the product of only two variable parameters: initial enrichment (х, %) and ratio of the burnt fuel mass (approximately equal to the mass of fission products) to the initial mass of fissionable nuclides (that is, in the loaded fresh fuel), ∆M_{f} /M_{5}. By comparing expressions (5) and (2), we obtain an important relation
(∆M_{f} /M_{5})(n + 1)/n = В∞ (m_{f} /E_{f})/x (6)
By definition, the righthand part in the above expression does not depend on the refueling multiplicity. Therefore, the lefthand part is not expected to depend as well on n, that is, the relative mass of fission products in extracted fuel during refueling depends only on the refuel in result unknown before. Meanwhile, as shown in Fig.
Dependence of the product of the relative mass of fission products (burntup nuclides, ΔM_{f} /M_{5}) by the function of refueling ratio, (n+1)/n with n = N_{CORE}/N, on enrichment of replaced FAs. Source: plotted by authors based on grid diagram data in
∆M_{f} /M_{5} = 1.53n/(n + 1) (7)
As it follows from Fig.
Finally, it follows from expression (7) with typical values of n=3–5 that ∆M_{f} /M_{5}=1.1–1.3, that is the mass of fissionable nuclides (mass of fission products) burnt exceeds by 10 to 30% the initial mass of uranium235 in fresh fuel due to the generated plutonium burnup which is confirmed by experimental data
Influence of uranium fuel enrichment and burnup on the mass of fission products in the PWR reactor SNF with one fourth of the reactor core refueled. Source: authors’ calculation of the ∆M_{f}/M_{5} values using formula (4) and of the (∆M_{f} /M_{5})(n + 1)/n value using formula (7) based on data in
Average uranium235 enrichment of replaced fuel x, %  Average burnup of replaced fuel В,MW·day/kgU  Relative mass of fission products in replaced fuel, ∆M_{f}/M_{T}, kg/t h.m.  Ratio of fission product mass in SNF to U235 mass in fresh fuel ∆M_{f}/M5  Parameter (∆M_{f}/M_{5})(n + 1)/n 

3.8  44.9  47  1.23  1.54 
4.5  54.3  57  1.27  1.59 
5.4  64.1  67  1.24  1.55 
6.5  73.8  78  1.2  1.50 
7.5  84.0  89  1.19  1.49 
8.5  93.7  99  1.16  1.45 
Substituting the obtained relation (7) into expression (5), taking into account (1) and (2), leads to the soughtafter analytical relationship of fuel burnup with fuel enrichment, refueling ratio, thermal stress and reactor campaign in the following form
B (MW×day/kgU) = 14.8x (%)n/(n + 1); В∞ = 14.8x (%). (8)
B (MW×day/kgU) = 14.8x (%) – q (kW/kgU)×T (day)/1000. (9)
Burnup calculations using formulas (8) and (9) describe satisfactorily the grid diagrams contained in
Relationship between average burnup (В, MW·day/kgU), enrichment (х, %) and refueling multiplicity (n≥1) for uranium fuel for different thermal neutron reactors. Source: plotted by authors based on experimental and calculated data in
As it follows from expression (9) and Fig.
Therefore, analytical expressions (7)–(9), obtained for the first time in the paper, allow estimating analytically the dependence of nuclear fuel burnup on fuel enrichment, refueling multiplicity(or number of discharged FAs), reactor campaign(refueling interval) and fuel thermal stress, which is required for plotting grid diagrams as a convenient tool for selection of the fuel cycle parameters.
The fuel component, Y_{F}, (Rub/kW·h) in the NPP operating costs includes the FA fabrication and spent nuclear fuel (SNF) handling cost and is proportional to the reactor fuel demand, Р (kg/year)
Y _{Т} = P (C_{FA} + C_{SNF}) = PC_{NFC}. (10)
Quantity С_{NFC} = С_{FA} + С_{SNF} can be called as the cost of the nuclear fuel cycle (open or closed) in terms of 1 kg of uranium (or heavy metals) in fuel (Rub/kg h.m.), including the FA cost, С_{FA} = С_{Х} + С_{FAB}, and the SNF handling cost, С_{SNF}. Quantities С_{Х} and С_{FAB} are the costs of enriched uranium and FA fabrication; Р is the annual average reactor fuel demand (kg/year) defined by the ratio of the annual average thermal power of the reactor, Q (MW), to the average fuel burnup, В, (MW·day/kg) according to (1):
P = 365×Q/B = E/(24ηB), (11)
where E = W·∆t·ICUF is the annual average amount of the electricity sold (MW·h/year); W is the installed electric power of the NPP unit, MW; Δt is the number of hours per year (8760 h/year); and Q=W∙ICUF/η is the reactor thermal power with gross efficiency η. In expressions (11), numerical coefficients 365 and 24 take into account the number of days per year and the number of hours per day in accordance with the commonly accepted dimensionalities of the initial quantities. Thus, for current PWRtype reactors with typical parameters such as W = 1200 MW, η = 34%, В = 55 MW·day/kgU, and ICUF = 0.85, the annual fuel demand is Р ≈ 20 t/year.
The ratio of fuel costs to the electricity sold, Y_{F}/E, represents the fuel component of the electricity cost (Levelized Cost of Electricity or LCOE) (
LC _{F} = Y_{f} /E = P (C_{FA} + C_{SNF})/E = (C_{FA} + C_{SNF})/(24ηB). (12)
If burnup is measured in MW·day/kgU, then LC_{F} is expressed in Rub/MW·h.
To estimate the cost of manufacturing the FAs replaced with fuel mass Р (kg/year) and enrichment х is necessary to know the consumed mass of natural uranium, F (kg/year) and uranium isotope separation work, R (SWU/year), found by standard expressions (
F = P (x – y)/(c – y); R = PΦ(x) + DΦ(y) – FΦ(c); (13)
Φ(z) = (1 – 2z)ln(1/z – 1); z = x, y, c, (14)
where D = (F – P) is the mass of depleted uranium with the mass concentration of uranium235, у; с = 0.711% is the mass concentration of uranium235 in natural uranium; and Φ(z) is the separation potential. As a result, for the production cost of fuel assemblies (per 1 kg of h.m.), including the costs of purchasing natural uranium and its conversion into uranium hexafluoride, uranium isotope separation, waste disposal and fuel assembly fabrication, we obtain the expression
C _{FA} = C_{F} (x – y)/(c – y) + C_{R}[Φ(x) + Φ(y)(x – c)/(c – y) –
– Φ(c)(x – y)/(c – y)] + C_{D} (x – c)/(c – y) + C_{FAB}, (15)
in which C_{F} = C_{U3O8} +C_{UF6} is the price of natural uranium hexafluoride; C_{U3О8}, C_{UF6}, C_{D} are prices of natural uranium (in the form of oxide concentrate), oxide concentrate conversion to uranium hexafluoride, and recycling of depleted uranium hexafluoride (waste) per 1 kg of uranium metal (Rub/kg); and C_{R} is the separation work price (Rub/SWU). It follows from expression (15) that the FA cost with the preset fuel enrichment level, х, and determined prices (C_{F}, C_{R}, C_{D}, C_{FAB}) depends only on the separation waste dump depth (tails assay), у, while there is an optimal tails assay, у_{0}, with which the FA cost is as small as possible. Quantity у_{0} depends only on the ratio of prices (C_{F} + C_{D})/C_{R} (
Historical dynamics of market prices for natural uranium and uranium conversion/enrichment services, and of estimated cost of enriched uranium (х = 4.95%). Source: compiled by authors based on data in
Parameter  2011  2018  2021 

Uranium oxide concentrate price, С_{U3O8}, $/kgU  148  65  91 
С _{UF6}conversion price, $/kgU  11  10  19 
Uranium hexafluoride price, C_{F}, $/kgU  159  75  110 
Separation work price, C_{R}, $/SWU  149  36  55 
Optimal tails assay, у_{0}, %  0.220  0.155  0.158 
Enriched uranium cost, С_{x}, $/kgU  2772  1002  1496 
The costs of fuel assembly fabrication and SNF management are to a lesser extent determined by market quotations, but may depend on the depth of fuel burnup (enrichment). Since according to data in
Influence of uranium fuel burnup on the NFC cost characteristics. Source: compiled by authors based on data in
Burnup depth, В, MW·day/kgU  45  55  65  75  85  95  
FA fabrication cost, С_{FAB}, $/kgU  300  330  360  390  420  450  
SNF transportation cost, С_{TR,} $/kgU  230  280  330  380  430  480  
SNF encapsulation and disposal cost, С_{DIS}, $/kgU  610  745  880  1015  1150  1290  
SNF handling cost, С_{SNF}=С_{TR}+С_{DIS}, $/kgU  840  1025  1210  1395  1580  1770  
Fuel enrichment*,х, %  3.9  4.6  5.6  6.5  7.3  8.2  
Enriched uranium cost**, С_{x}, $/kgU  2011  2090  2545  3200  3800  4330  4940 
2018  760  920  1150  1360  1540  1750  
FA cost, С_{FA}=С_{x}+С_{FAB}, $/kgU  2011  2390  2875  3560  4190  4750  5390 
2018  1060  1250  1510  1750  1960  2200  
Fuel component of NPP electricity cost***, LC_{F}, $/MWт·h  2011  8.8  8.7  9.0  9.1  9.1  9.2 
2018  5.2  5.1  5.1  5.1  5.1  5.1 
It is important to note that, according to the presented results, the market prices for natural uranium and separation work have major effect on the fuel component of the electricity cost as compared with effects from fuel burnup and, accordingly, fuel enrichment.
The paper presents newly obtained analytical expressions (8), (9) for estimating the burnup of nuclear fuel depending on fuel enrichment, refueling periodicity, reactor campaign and specific thermal stress of fuel in a wide range of these parameters (without taking into account the constraints from the physicochemical processes in conditions of high burnups) as applied to thermal reactors. It has been shown that nuclear fuel burnup is directly proportional to the multiplication of only two parameters: enrichment of the fuel loaded during refueling, and ratio of the burnt fuel mass (≈ fission product mass) to the mass of fissionable nuclides in the loaded fuel according to expression (4). The latter ratio (∆M_{f} /M_{5}) depends practically only on the refueling multiplicity and changes in a range of 1.0 to 1.5. The obtained analytical expressions for the burnup estimation are convenient as applied to variablebased economic, thermalphysical, strength and other calculations for reactor fuel batches with different cycle durations (12 to 24 months) and fuel enrichments (up to 10%), including the accident tolerant fuel under development.
It has been shown that fuel burnup increases practically linearly with the enrichment growth in the considered range of 0.7 to 10% with the preset refueling multiplicity, as shown by expression (8), and decreases linearly with the reactor campaign(refueling interval) increase with the preset fuel enrichment according to (9).
It has also been shown that the fuel component of the PWR NPP electricity cost is not so particularly sensitive to the fuel burnup change but is much more sensitive to the volatility of market prices for natural uranium, conversion and separation work or to changes of enriched uranium cost.