3.6 Secondary analyses
4.1.2 Fuel composition
This section compares the Ants-based nuclide inventory to Serpent. First, the accuracy of the micro-depletion method is assesed on a general level by comparing the nuclide inventory at the end of irradiation. Second, a few selected nuclides are compared as a function of burnup for a more detailed analysis.
General statistics
The irradiated fuel in the reference solution contained 1436 different nuclides. Ants solution contained additional 25 nuclides, but their quantities were zero or extremely small and they were ignored. In addition, nuclides in the reference solution with an average number density lower than one nuclide per cubic meter were discarded from the analysis.
The scatter plot in figure 4.3shows the average number density of each nuclide calculated with Ants and the relative difference compared to Serpent. The bulk of all nuclides have a relative difference between 10−4 and 10−2 but there are few notable outliers. These outliers are overestimated and they are light nuclides 3H, 3He, and
7Li. Irradiation histories have otherwise similar point clouds but the bulk of soft spectrum points is slightly above nominal and hard spectrum points.
Table 4.1 shows the cumulative number of nuclides as a function of relative difference. For all irradiation histories, approximately 90 % of all nuclides have a relative difference smaller than 0.01. Furthermore, approximately half of all nuclides
100 103 106 109 1012 1015 1018 1021 1024 1027 1030 Average number density ¯N m13
10−7
10−6 10−5 10−4 10−3 10−2 10−1 100 101
Relativedifference
3H
3He
7Li Soft
Hard Nominal
Figure 4.3: Nuclide average number density calculated with Ants and the relative difference compared to Serpent in a 2D single assembly depletion problem. Data point color indicates the depletion history and the size is proportional to the logarithm of the average number density. Three outliers at the top of the figure are labeled with their names.
Table 4.1: Cumulative number of all nuclides with a lower relative difference compared to the reference solution. Columns indicate the number of nuclides with a lower relative difference than the value at the top row.
Relative difference < 10−4 < 10−3 < 10−2 < 10−1 < 100 < 101
Nominal # 113 675 1183 1317 1347 1350
% 8.4 50.0 87.6 97.6 99.8 100
Soft # 14 137 1211 1330 1346 1349
% 1.0 10.2 89.8 98.6 99.8 100
Hard # 72 699 1238 1323 1349 1352
% 5.3 51.7 91.6 97.9 99.8 100
from nominal and hard histories have less than 0.001 relative difference compared to Serpent. The irradiation history with soft spectrum has only 10 % of all nuclides in this category. Only thirty or twenty nuclides have a relative difference greater than 10 %.
So far, compared differences have been in absolute values. To find out if nuclide numberdensities are either over- or underestimated, results have to be categorized concerning their sign. Table4.2shows the relative frequency of the relative difference.
The bottom row of the table shows that the majority of all nuclides are underestimated.
47
Table 4.2: Relative frequency of Ants-based nuclide average number density relative differ- ence. The results of each depletion history are divided into three columns. Overestimated values are labeled with a plus sign, underestimated values are labeled with a negative sign, and the sum is labeled with a sum symbol. The last row also shows the sum over the whole column. All values are in percentages.
Nominal Soft Hard
Relative difference + − ∑︁ + − ∑︁ + − ∑︁
0 ... 10−4 3.8 4.6 8.4 0.6 0.4 1.0 2.8 2.5 5.3
10−4 ... 10−3 12.8 28.8 41.6 5.6 3.5 9.1 6.4 40.0 46.4 10−3 ... 10−2 2.4 35.3 37.6 4.1 75.5 79.6 2.7 37.2 39.9 10−2 ... 10−1 0.2 9.7 9.9 2.4 6.4 8.8 1.3 5.0 6.3
10−1 ... 0.2 2.2 2.4 0.5 0.9 1.4 0.3 1.8 2.1
∑︁ 19.4 80.6 100.0 13.2 86.8 100.0 13.4 86.6 100.0 For nominal history, the fraction of underestimated isotopes is 81 % and for soft and hard histories it is 87 %. Bulk of all nuclides have relative difference from 10−4 to 10−2, as was already seen from the figure4.3. Distributions for nominal and hard histories are rather similar, but the soft history is slightly exceptional. 76 % of all nuclides in the soft irradiation history are underestimated within a range 10−3 to 10−2 while corresponding values for nominal and hard histories are 35 and 37 % respectively.
To conclude, Ants calculates spent fuel nuclide inventories accurately compared to Serpent in the 2D single assembly depletion problem. The relative difference is less than 1 % for the majority of nuclides. Furthermore, one-order higher accuracy is obtained for half of all nuclides for nominal and hard histories. Even though the majority of nuclides are well predicted, few nuclides have a remarkable difference compared to the reference. Light nuclides 3H, 3He, and 7Li are miscalculated. In addition, numerous other unidentified nuclides have remarkable differences.
The soft depletion history has a larger relative difference likely due to the un- derestimation of thermal neutron flux. Figure4.2 shows that the relative difference of thermal flux is higher for soft history than the two other histories. The under- estimation of thermal flux leads to a lower fission rate and slower accumulation of fission products. On the other hand, fission products represent a large number of all nuclides explaining the trend of underestimating nuclide number densities. However, the impact may be the opposite for some nuclides like 235U that is depleted at a slower rate.
In addition to flux, few other factors may cause inconsistencies in results. First of all, errors in transmutation chains or bugs in software may lead to false results.
Second, the statistical error in both the homogenization process and the calculation of reference results causes a random error. Third, the two-group approximation may lead to errors. The miscalculation of light isotopes3H, 3He, and 7Li could be explained with some of these three factors but it was not investigated further in this thesis.
Nuclides born from high-threshold reactions are especially affected by bad statis-
tics. The reason is twofold. First, they are scored in the Monte Carlo simulation only if the energy of the incident neutron is above the threshold. Thus, the reference solution calculated with Serpent may have high statistical uncertainty. Second, the homogenized microscopic cross section is also affected by bad statistics. Therefore, both Serpent and Ants-based solutions may be far from reality, and comparing them will lead to inconsistent results.
Selected nuclides
Next, a few distinct nuclides are compared in detail to the Serpent reference solution.
The number density and the relative difference of 235U, 239Pu, 238Pu, 241Am,137Cs,
131I, 149Pm, and 149Sm is shown as a function of burnup. Results for each nuclide are compared for all three different depletion histories.
During irradiation, the inventory of 235U is depleted and it is partially replaced by239Pu as shown in figure 4.4. The fissile uranium content is depleted faster with the soft spectrum and more plutonium is accumulated with the hard spectrum. Ants accurately models this behavior and the relative difference for both nuclides is at most 0.7 % compared to Serpent. The content of 235U is overestimated and it is due to the underestimation of thermal flux. This leads to lower fission and neutron capture rate, thus the nuclide is depleted at a slower rate. Contrary to the fissile isotope of uranium,239Pu content is simultaneously accumulated and depleted. In this case, the net accumulation rate is faster compared to Serpent for all depletion histories.
Figure 4.5 shows that Ants can also predict the inventory of 238Pu and 241Am accurately in the two-dimensional depletion problem. These two minor actinides are
0 10 20 30
Burnup
MWd kgU
0.4
0.9 1.4 1.9 2.4
Averagenumberdensity¯N1 m3 ×1026 235U
Nominal Soft Hard
0 10 20 30
Burnup
MWd kgU
0.0
0.7 1.4 2.1 2.8
3.5 ×1025 239Pu
Nominal Soft Hard
0.0 0.2 0.4 0.6 0.8
−0.1 0.0 0.1 0.2 0.3 0.4
Relativedifference(%)
Figure 4.4: The evolution of two major fissile isotopes in light water reactors as a function of burnup. Ants results are drawn with circles and Serpent results with solid lines. The relative difference between Ants and Serpent is marked with crosses.
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0 10 20 30
Burnup
MWd kgU
0.00
2.25 4.50 6.75 9.00
Averagenumberdensity¯N1 m3 ×1023 238Pu
Nominal Soft Hard
0 10 20 30
Burnup
MWd kgU
0.0
0.5 1.0 1.5
2.0 ×1023 241Am
Nominal Soft Hard
−0.70
−0.35 0.00 0.35 0.70
−0.30
−0.15 0.00 0.15 0.30
Relativedifference(%)
Figure 4.5: The evolution of two minor actinides as a function of burnup. Ants results are drawn with circles and Serpent results with solid lines. The relative difference between Ants and Serpent is marked with crosses.
less abundant in spent fuel and they are formed from more complicated transmutation chains. 238Pu transmutation chains starts from neutron capture of 235U. This is reflected in underestimating the inventory at higher burnups. 241Am, on the other hand, is formed from the same chain as239Pu with two additional neutron captures and consequent beta decay. The relative difference follows asymptotically the same
0 10 20 30
Burnup
MWd kgU
0.000
0.375 0.750 1.125 1.500
Averagenumberdensity¯N1 m3 ×1025 137Cs
Nominal Soft Hard
0 10 20 30
Burnup
MWd kgU
0.0
0.2 0.4 0.6 0.8 1.0
×1023 131I
Nominal Soft
−0.4 Hard
−0.3
−0.2
−0.1 0.0
−0.5
−0.4
−0.3
−0.2
−0.1 0.0
Relativedifference(%)
Figure 4.6: The evolution of 137Cs and 131I as a function of burnup. Ants results are drawn with circles and Serpent results with solid lines. The relative difference between Ants and Serpent is marked with crosses.
pattern as 239Pu, even though initially the net effect under or overestimating the content is less clear.
Figure 4.6 shows the inventory of 137Cs and 131I as a function of burnup. Both nuclides have large fission yields and they are removed via beta decay with a half- life of 30 years and 8 days respectively. 137Cs accumulates linearly as a function of burnup and131I reaches equilibrium soon after startup. Ants predicts both inventories accurately even though they are slightly underestimated.
0 10 20 30
Burnup
MWd kgU
0.0
0.5 1.0 1.5
Averagenumberdensity¯N1 m3 ×1022 149Pm
Nominal Soft Hard
0 10 20 30
Burnup
MWd kgU
0.0
0.6 1.2 1.8 2.4
×1022 149Sm
Nominal Soft
−0.6 Hard
−0.4
−0.2 0.0
−0.2 0.0 0.2 0.4 0.6
Relativedifference(%)
Figure 4.7: The evolution of149Sm and the precursor149Pm as a function of burnup. Ants results are drawn with circles and Serpent results with solid lines. The relative difference between Ants and Serpent is marked with crosses.
Reactor poison 149Sm has a small direct fission yield and it is produced from the decay of 149Pm. Figure 4.7 shows the inventory for both nuclides. The half-life of
149Pm is 53 hours and it reaches equilibrium fast. Like 137Cs and 131I, its quantity is underestimated. However, the removal rate of149Sm is sensitive to thermal flux due to the high thermal neutron capture cross section. Therefore the content of 149Sm is slightly overestimated even though the production rate from the decay of149Pm is underestimated.
At the end of irradiation, all selected eight nuclides have relative differences smaller than one percent. The average number densities of selected nuclides and their relative differences are shown in table4.3. Consequent steps in the calculation chain use the composition at the end of irradiation.
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Table 4.3: Average number densities of eight different nuclides at the end of 2D assembly irradiation. All values are tabulated at a burnup of 34 MWdkgU. The first column with numeric values shows the average number density calculated with Ants and the second column shows the average number density calculated with Serpent. The last column presents the relative difference between the two.
Nuclide History Ants N¯ (︂ 1
m3
)︂ SerpentN¯ (︂ 1
m3
)︂ Difference (%)
235U Nominal 5.641·1025 5.620·1025 0.37 Soft 5.387·1025 5.353·1025 0.64 Hard 5.954·1025 5.936·1025 0.32
239Pu Nominal 3.172·1025 3.165·1025 0.23 Soft 3.016·1025 3.007·1025 0.29 Hard 3.398·1025 3.391·1025 0.21
238Pu Nominal 8.139·1023 8.162·1023 −0.28 Soft 7.702·1023 7.746·1023 −0.56 Hard 8.679·1023 8.695·1023 −0.19
241Am Nominal 1.698·1023 1.697·1023 0.10 Soft 1.574·1023 1.572·1023 0.10 Hard 1.859·1023 1.855·1023 0.18
137Cs Nominal 1.378·1025 1.380·1025 −0.15 Soft 1.376·1025 1.380·1025 −0.26 Hard 1.378·1025 1.380·1025 −0.15
131I Nominal 1.075·1023 1.076·1023 −0.01 Soft 1.071·1023 1.074·1023 −0.25 Hard 1.078·1023 1.078·1023 −0.04
149Pm Nominal 1.601·1022 1.601·1022 −0.02 Soft 1.602·1022 1.608·1022 −0.35 Hard 1.595·1022 1.597·1022 −0.11
149Sm Nominal 2.070·1022 2.063·1022 0.31 Soft 1.909·1022 1.903·1022 0.34 Hard 2.295·1022 2.291·1022 0.17