On the problems of reusing reprocessed uranium by enrichment in schemes based on ordinary cascades

The problem of spent nuclear fuel attracts considerable attention while its quantity is accumulating worldwide. The problem of long-term supply of the fresh fuel also remains important. One of the strategies to solve both problems is reusing the spent nuclear material. The uranium, in this way, could be recycled multiple times in light-water reactors. In order to recycle the uranium, it is extracted from the irradiated fuel during the reprocessing and then enriched in 235U, taking into account the limitations on reactor-born isotopes 232,236U in the final product. The only way to do this is enrichment in cascades of gas centrifuges. However, not every cascade scheme is able to re-enrich the uranium for multiple recycles, utilizing the whole amount of uranium extracted from the irradiated fuel each time. This study shows that configurations based on ordinary three-flow cascades could not be used for this purpose. In particular, we have shown that starting from the second uranium fuel cycle, such schemes are no longer able to reclaim the necessary proportion of the reprocessed uranium.


Introduction
The risks and costs of storing the growing inventory of spent fuel will continue to increase; and in the absence of an end point, it will eventually become a significant burden [1]. The limitation of natural uranium is also an obstacle for nuclear energy because, for today, only about 8 megatonnes can be extracted with the price for 1 kg not higher than 260$ [2]. Even the fact that the worldwide amount of uranium resources on the globe is estimated as ≈58 megatonnes, (where about 80% are unconventional resources) impose long-term constraint on the nuclear industry [1].
Spent fuel recycling could be one of the options to avoid these negative effects due to these properties of the reprocessed uranium (RepU): (i) Uranium recovered from the spent nuclear fuel (SNF) makes up the majority of its content; (ii) RepU contains more fissile 235 U than natural one.
Thus, spent fuel reprocessing could provide additional fissile resource for nuclear power plants and decrease the amount of high-level waste. However, the spent fuel reprocessing is difficult, so not many countries have production capacities to do it. Even among global nuclear leaders, only a few countries, such as Russia, France, Japan, decided to reprocess the SNF to close the nuclear fuel cycle.
The next operation to return uranium to the fuel cycle is to produce low-enriched uranium (LEU), which could be used in a light-water reactor (LWR) -the type of power reactor that dominates worldwide. As we need to raise the level of 235 U, the possibility of reusing the uranium depends on separation technology -cascades of gas centrifuges [3,4]. The major restrictions here are caused by reactor-born 232,236 U and an order of magnitude higher concentration of 234 U than in the NatU. The 236 U acts as a parasitic neutron absorber that inhibits a chain reaction and should be compensated according to equation 1, where C 235eq. stands for the equivalent of 235 U concentration needed to be reached in the final product, with the surplus of 235 U, which depends on 236 U presence (f (C P roduct 236 )), to the initially required concentration of 235 U in LEU product, that is C P roduct 235 [5]. In our case f (C P roduct 236 ) = RCC × C P 236 , where RCC equals to 0.29.
The 232 U is a particularly dangerous source of radiation pollution due to intense gamma radiation (2.6 MeV) emitted by a short-lived daughter 208 Tl [6]. 232,234 U add detrimental alpha particles to the uranium hexafluoride gas used in the uranium enrichment process [7]. As a result, the international specifications, such as an ASTM, restrict the amount of 232,234,236 U in commercial LEU. And as these isotopes are lighter than the prevailing 238 U, they tend to accumulate at the same side of the cascade as the desired 235 U, when the abundant 238 U goes to the other side.
This problem makes it difficult to use an ordinary three-flow cascade for uranium enrichment (figure 1) -a cascade of gas centrifuges (triangle in figure 1), consisting of several stages to achieve the required concentration of 235 U in the LEU product [8,9]. This cascade has 3 external flows: 1) Feed -mixture input to be separated into the cascade; 2) Product --the flow enriched in 235 U; 3) Waste -the flow depleted in 235 U.  In the general case, such a cascade does not allow enrichment of reprocessed uranium with a simultaneous correction of its isotopic composition -the concentrations of even-numbered uranium isotopes [10]. An exception may be compositions with a relatively low initial content of even isotopes, which does not fit for modern light-water reactors.
To make use of the reprocessed uranium, the necessary modification of the ordinary cascade (figure 1) has been proposed [5,11,12]. Such schemes are based on preliminary or subsequent RepU admixing -to the feed or product flow, or final dilution of enriched RepU [13]. These However, to reach sustainable reuse of fissile materials and to avoid accumulation of the irradiated nuclear fuel, it is reasonable to produce the same amount of reactor-quality LEU as the amount of used fuel allocated for this operation. In this way, as the uranium makes up ≈93% of SNF, each kilogram of fresh LEU should be produced from ≈0.93 kg of the RepU to be utilized [14].
This article evaluates the applicability of the simplest modifications of an ordinary cascade to solve the problem of repeated uranium recycling. In other words, the article investigates mass transfer in cascades based on ordinary cascade designed for enrichment of reprocessed uranium in order to assess if they are capable to solve the problem of returning the whole amount of reprocessed uranium to the nuclear fuel cycle in the regime of multiple recycling when the isotope composition is degrading each cycle. These schemes have additional input of natural uranium to dilute unwanted 232 U to meet standard fresh LEU requirements [15]. At the same time, the reasons for the emergence of difficulties with the enrichment of reprocessed uranium with the simultaneous fulfillment of the conditions for even-numbered isotopes and the 1:1 use of the RepU using such schemes are revealed. Then, a method for a priori assessment of the cascade's ability to utilize the entire amount of irradiated uranium is described. In the next section, the simplest schemes for enrichment of reprocessed uranium will be analyzed.

Materials and methods
Let us consider each of the proposed modifications of an ordinary cascade for enrichment of RepU separately.

Examined schemes
Scheme 1 (figure 2) is one of the simplest possible modification of ordinary cascade which can be used to enrich reprocessed uranium. It enriches RepU in 235 U to a higher level than it is required in the final product, and this intermediate product -P 0 is then blended with NatU (or the prepared in advance LEU) to dilute the 232 U concentration. The W 0 flow is the depleted uranium (DepU). The dilution is carried out as follows: enriched uranium produced from the RepU in the cascade, is diluted with a mixture that does not contain minor isotopes, achieving the desired content of the 235 U isotope in the final LEU product.
In the Scheme 2 (figure 3) NatU, enriched to a higher level than necessary, is then mixed with RepU. This configuration avoids contamination of the separation equipment by 232 U from RepU.
For all variants (figures 2-4), the ratio between the consumption of the reprocessed and the diluent of natural origin is determined by the limit of the permissible concentration of 232 U in the final product -low-enriched uranium. The negative reactivity of 236 U compensation should also be taken into account. At the same time, the concentration of 235 U should not be lower than that required for LEU with certain properties.

Statement of the problem
For the considered schemes, a series of computational experiments have been conducted. As a math model to calculate the enrichment procedure, we employed the R-cascade model also known as Matched Abundance Ratio Cascade [16,17,18]. Uranium hexafluoride gas is a working substance in the uranium enrichment performed by the gas centrifuge [19,20]. The production of fresh LEU fuel from RepU and the additional 235 U source was simulated for the following conditions:    (i) Reprocessed uranium is derived from the light-water energy reactor. As an example, we will consider the isotopic composition of the reprocessed uranium extracted and recovered from the reactor of Russian design -VVER. The original isotopic composition of the RepU from the second recycle, presented in the table 1 [21]. (ii) Key components for which non-mixing in R-cascade model is set are: 235 U and 238 U. (iii) Separation factor is equal to 1.2 for 235 U F 6 to 238 U F 6 [22]. (iv) The required concentration in the final product is 4.95%, which is typical for light-water reactors. (v) Consumption of reprocessed uranium per unit of the final LEU product: 0.93 kg per 1 kg of LEU [23]. (vi) By default, the concentration of 235 U in W 0 equal 0.1% [24]. (vii) 234 U to 235 U ratio should not exceed 0.02 [23]. (viii) The reactivity compensation ratio to neutralize the undesirable effect of neutron capture by isotope 236 U is 0.29 and also corresponds to Russian light-water reactors [24]. (ix) The concentration of 232 U in LEU is limited by 5·10 −7 % [24].
With this representative formulation of the problem, we will check if the enrichment schemes based on an ordinary cascade could solve the uranium recycling problem.

Results and discussion
Let us start the analysis with the Scheme 1, when the RepU of the second recycle is enriched to a level exceeding the required concentration of 235 U, and then diluted with NatU ( figure 2).
The LEU product must simultaneously correspond to the required 235 U enrichment level, 232 U limitation and 236 U compensation condition. While the problem statement is associated with 3 concentrations and RepU P ratio, there are only two control parameters in such a problem: the 235 U output concentration and the proportion of mixed flows. Therefore, the possibility of obtaining a solution to the problem under such conditions is not defined. It is important to note that although the output concentration of 235 U is a control parameter, its change inevitably entails a change in the concentrations 232,236 U, thereby introducing additional uncertainty to the problem.
The curves on figures 5-8 reflect the dependencies of absolute values of residual errors δ 1 ,δ 2 in solving the system of equations for cascade from the proportion of natural uranium to the pre-enriched RepU. Here, δ 1 = C P 235 − C P n + RCC × C P 236 and δ 2 = C P 232 − 5 × 10 −7 × 10 5 . Figure 5. 235 U and 232 U discrepancies for C 235 P 0 = 15% Figure 6. 235 U and 232 U discrepancies for C 235 P 0 = 30% Figure 7. 235 U and 232 U discrepancies for C 235 P 0 = 50% Figure 8. 235 U and 232 U discrepancies for C 235 P 0 = 65% According to its physical meaning, the value of δ 1 is the absolute deviation of the concentration of 235 U isotope (expressed in mass fractions) in the final product (after mixing) from the required value, taking into account the compensation of 236 U, and the value of δ 2 is the difference between the actual concentration of 232 U in the final product and the required value in compliance with the accepted limitation. In order to compare the indicated values in one figure, δ 2 is multiplied by a specially selected numerical coefficient. δ 1 and δ 2 correspond to discrepancies in the solution of a system of nonlinear equations, which, in the case of a successful finding of a solution to the system of equations (δ 1 = 0, δ 2 = 0), will be equal to a value that does not exceed the maximum permissible calculation error in the numerical solution. Figures  5-8 are plotted for various values of the 235 U concentration in the pre-enriched RepU. For the successful solution of the system of nonlinear equations, both functions must equal zero (δ 1 = 0, δ 2 = 0) for the same value of the argument, which cannot be achieved, which is reflected in the given figures. Thus, the results obtained show the impossibility to use Scheme 1 (figure 2) with natural uranium as a diluent for the case of multiple uranium recycling.
To expand the possibility of solving this problem using a scheme based on an ordinary cascade, when the RepU is pre-enriched, it is necessary to replace the diluent with pre-enriched uranium that does not contain minor isotopes. In this case, it will be possible to find a solution when the conditions for 235 U and 232 U are simultaneously fulfilled (to achieve the equality of both residuals to zero). To investigate the range of parameters, where the cascade can operate and where it is most effective, two main quantities will be investigated below: the concentration of 235 U in the output flows P 0 and W 0 . This will help illustrate, how the shares of NatU and RepU in the final LEU product change and how the separative work behaves. Figure 9 illustrates that the Scheme 1 (figure 2) consumes the NatU to produce the necessary LEU product, at the level close to the ordinary cascade without RepU to produce the fresh LEU equivalent. For LEU from NatU the ratios would be 7.41, 8.55 and 11.3 for C 235 W 0 =0.05%, 0.15% and 0.3% respectively. As we could see, the lower right corner of the plot 9, where the 235 U from RepU is raised to the highest levels (about 50% in mixture) and the feed is depleted greater, corresponds to the best configuration in terms of NatU savings. Having the higher level of 235 U extraction from RepU with lower 235 U concentration in W 0 helps us save the NatU, allowing for LEU diluent of lower 235 U level in figure 10 for Scheme 1 (figure 2), and having the deeper depleted tails (C 235 W 0 =0.05%) allows to save some separative work as shown in figure 11 due to the more efficient extraction of 235 U from the RepU -a mixture in which the fraction of this target isotope is initially greater.  But, as figure 12 shows, there is no way to fulfill the conditions for returning a given proportion of RepU per unit of product. And the proportion of recycled material used per unit of product changes only slightly with an increase in the 235 U content in P 0 . That is, the lengthening of the enrichment part of the cascade does not lead to a significant increase in the RepU reclaim efficiency, therefore there is a little reason to increase the concentration of 235 U in P 0 higher than 20%. The merging of the curves in figures 11 and 12, corresponding to different concentrations of 235 U in the enriched RepU, is explained by the equivalence of the 235 U masses in each of these streams, which reflects the equivalence of the contribution from the enriched RepU to the formation of the final product.  It is possible to illustrate the costs of the separative work from the components of the cascade scheme using figure 13. To assess the impacts on separative work from different ordinary cascades in the scheme, as all separation devices in the R-cascade work in the same operating conditions, we could compare the proportions of centrifuges for preparing a diluent from natural uranium to centrifuges used for preliminary RepU enrichment. From that we could see that the NatU enrichment part requires more separative work and its share in total consumption decreases with a decrease in the 235 U content in W 0 , which is explained by higher level of 235 U involvement from RepU.  Thus, the analysis of figures 9-12 shows the unsuitability of Scheme 1 (figure 2) with dilution of pre-enriched RepU with a mixture that does not contain minor isotopes to solve the problem under conditions of multiple recycling, since using such a scheme it is impossible to involve a given amount of RepU (0.93) in the production of a unit of commercial LEU. At the same time, an analysis of the graphs demonstrates that a decrease in the 235 U concentration in W 0 makes it possible to achieve the best implementation of the Scheme 1 shown in 10: to achieve savings in natural uranium per unit of product 9, to get by with a lower concentration of 235 U in the LEU diluent (figure 10), and also to save the separative work ( figure 11), while the consumption of RepU per unit of product will be less by an insignificant amount ( figure 12), not exceeding 1% in comparison with the cascade for enrichment of natural uranium. An analysis of the plots (figures 9, 10) demonstrates the possibility, with an increase in the level of enrichment of the RepU, to obtain greater savings in natural uranium due to the lower required concentration of 235 U in the LEU diluent. However, it should be noted that exceeding the 235 U concentration of the LEU level (20%) may be unacceptable in separation process, since such material falls into the category of highly enriched uranium (HEU), the production of which is limited [25].
Let us move on to the analysis of the Scheme 2 (figure 3), where pre-enriched natural uranium is mixed with reprocessed uranium returned to the fuel cycle. In such a cascade, it is possible to obtain a single solution for each combination of parameters. In this configuration, as in the case of using the Scheme 1 (figure 2), the consumption of RepU per unit of LEU product is insufficient (0.75), and the level of natural uranium enrichment in the cascade reaches 16% for different 235 U concentrations in the waste flow of this cascade. Consequently, such modifications of the three-flow cascade are not adequate for the task.
Another option for the implementation of the simplest modification of an ordinary cascade used to enrich natural uranium is preliminary mixing of natural uranium with RepU before feeding the resulting mixture to the cascade in proportion, determined from the limitation on even-numbered isotopes in the final LEU product. This Scheme 3 is shown in figure 4. The analysis shows (figure 14) that it is also impossible to achieve the required proportion of the consumption of RepU per unit of the final product (the solutions correspond to the intersections of curves with gray lines). The upper horizontal axis here and in the adjacent plot ( figure 15) shows the proportion of the RepU in the feed isotopic mixture. Figure 15 shows that consumption of natural uranium per unit of product for the same set of found solutions could be even higher than in ordinary cascade for NatU enrichment.
Thus, the numerical experiments proved that we could not employ the cascades based on the ordinary three-flow cascade to enrich the degraded isotopic composition of reprocessed uranium. But what if we could assess the capability of the observed scheme to solve this task a priori? According to the balance equations 2 for material flows from which a special case was obtained by passing to the limit for the lightest isotope 3, which shows that to satisfy 1:1 RepU to LEU product requirement only when 232 U concentration in the initial RepU feed lower than the limitation of 232 U presence in the final product.
, where F is Feed, P is Product, and W is Waste flows. Using this equation, the maximum possible share of the feed flow in P containing 232 U can be calculated as the unknown variable of the equation, assuming that all the original 232 U will end up in the product 3: The same limitation for RepU P appears in figures 12 and 14, but 0.93 is required, from which it can be concluded that it is impossible to completely return (1:1) the RepU to the fuel cycle using schemes of the simplest modification of an ordinary cascade, because they not isolate the 232 U isotope, which should be withdrawn from the cascade at some point.
Let us calculate the limiting contribution of 235 U from the RepU, taking it as X: So, taking the RepU as 0.93 and P as 1, as in the problem statement, the limiting contribution of 235 U from RepU is limited to about 28%, which corresponds to the natural uranium savings. But in reality it would be even less due to the need to compensate for 236 U and the fact that some 235 U goes to the waste flow. As a result, as the 232 U in the RepU of the second cycle is already higher than the limitation, it is impossible to recycle the same amount of uranium multiple times with the ordinary-based cascades, either the RepU is preliminary enriched or used as a diluent of pre-enriched natural uranium. But there is a way out -to employ the cascades designed to detach the 232,234 U from 235 U. Such schemes are based on two linked cascades, where in the first one, the 235 U is enriched with a simultaneous increase in the concentration of 232,234 U isotopes, but in the second cascade, the isotopic mixture is divided into light 232,233,234 U and heavy 235,236 U groups [26]. But this segregation of the lightest isotopes is associated with the accumulation of toxic waste that could be diluted by depleted uranium and reemployed. So, out of this paper's scope, there are schemes that could succeed in the assigned task.

Conclusion
Considering the reprocessed uranium isotopic composition of the second recycle, the enrichment cascades based on an ordinary three-flow cascades could not be used to consume the whole amount of such material for fresh LEU production. We show that to recycle spent fuel multiply, the degrading uranium isotope composition should be assumed. It shows up in the growing presence of 232,234,236 U, which spoils the effect of their dilution by compositions without 232,236 U (and with lower 234 U concentration than in NatU). However, the outlined options could be used to re-enrich the reprocessed uranium on the first recycle at the required level of material involvement.