III. Forward Uncertainty Quantification of Spent Fuel
3.5. Uncertainty Quantification Results
3.5.2. Uncertainty Quantification due to Nuclear Data
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Table 3.9. Maximum RSD of surrogate point-wise predictions at 23 years.
Activity Decay heat Neutron source RSD (%) 3.14E-02 4.91E-02 1.86E-02
Gamma source Burnupa k-infa RSD (%) 8.79E-03 3.00E-04 2.15E-02
a See footnote of Table 3.5.
Table 3.10. Error of surrogate model mean and standard deviation at 23 years (%).
Activity Decay heat Neutron source RSD of mean 4.83E-03 7.06E-03 2.74E-02
RSD of σ 4.91E-01 6.56E-01 8.94E-01
Gamma source Burnupa k-infa RSD of mean 1.58E-03 7.95E-05 4.07E-03
RSD of σ 1.47E-01 9.89E-03 8.64E-01
a See footnote of Table 3.5.
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because its initial amount remains almost the same during burnup. Figure 3.11a shows the uncertainty of plutonium isotopes due to cross section. When small amount of 238Pu are produced at low burnup, 38Pu has large uncertainty. At discharge, there is 1.5% to 3.5% uncertainty in
238,239,240,241,242Pu. Figures 3.12a and 3.13a show the uncertainty in minor actinides caused by cross section. These values are higher than the major actinides uncertainties. During burnup, the uncertainty of 243Am and 244,245Cm gradually decreases, but remains high between 8% and 13% at the end of the irradiation. The decreasing uncertainties are due to the growing trend of minor actinide concentrations during burnup. From 2 to 37 GWd/tU burnup, the uncertainty of 241Am fluctuates between 2% and 3%. The inter-cycle cooling intervals modeled in the irradiation history induce the decline in 241Am uncertainty around 21 and 28 GWd/tU. The assembly was irradiated in the reactor in four cycles, as shown in Table 3.1. There were three down times during irradiation, which occurred at burnups of 11, 21, and 28 GWd/tU, respectively. As a result, around these burnup values, various uncertainties in Figures 3.10 – 3.15 may show a dramatic increase or drop.
Fluctuations can be eliminated by using constant power in the depletion. The constant power can be calculated by dividing the discharge burnup by the number of operating days. However, this approach can introduce another modeling uncertainty, namely, power history uncertainty, which can bias the nuclide concentrations of nuclides, such as 241Am which are sensitive to the power history [87]. Figures 3.10c – 3.13c show that the uncertainties coming from the fission yield are quite minimal and less than 1% for the major and minor actinides addressed thus far. The reference [60] mentions the same observation. Figure 3.14a depicts cross section uncertainties in certain fission products. Uncertainties for 135Cs and 154Eu are increasing, although they are still very low at 0.5% at the end of the burnup. During depletion, 137Cs and 144Ce have almost zero uncertainty.
134Cs, on the other hand, has a high uncertainty throughout irradiation, ranging from 4% to 5%.
Figure 3.14c shows that uncertainties in fission products caused by the fission yield are significant and larger than those in minor and major actinides. The uncertainty of radiation source terms caused by cross section during burnup are shown in Figure 3.15. With increasing burnup, the activity and gamma sources have lower uncertainties, whereas the decay heat and neutron source have higher uncertainties. These uncertainties, on the other hand, are small and remain below 0.6%
after irradiation.
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(a) (b)
(c)
Figure 3.10. Uncertainties of uranium isotopes during burnup caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.11. Uncertainties of plutonium isotopes during burnup caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.12. Uncertainties of minor actinides during burnup caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.13. Uncertainties of curium isotopes during burnup caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.14. Uncertainties of fission products during burnup caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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Figure 3.15. SNF source term uncertainties versus burnup, caused by cross section, fission yield and modeling parameters.
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Next, the cross section and fission yield uncertainties in nuclide concentrations and source terms after burnup are analyzed. Figures 3.16 – 3.21 show the uncertainties. Figure 3.16a shows the cross section uncertainty of uranium isotopes during cooling. Between 10 and 500 years of cooling, there is a gradual increase in 234U uncertainty. Although, up until 10,000 years, this uncertainty is less than 2%. Because the nuclide concentrations of 235,236,238U do not change after burnup due to their extremely long half-lives, their uncertainties remain constant. These major actinides have uncertainties of 1.2%, 1.1%, and 0.0%, respectively. Figure 3.17b shows the uncertainty of plutonium isotopes due to cross section during cooling. The uncertainties for
238,239,240,242Pu are essentially constant over 10,000 years and range between 2% and 3%. After 200 years, there is a significant increase in 241Pu uncertainty, which reaches around 13% after 500 years and remains constant from 500 to 10,000 years after discharge. To investigate the origin of the dramatic increase in 241Pu uncertainty, we refined the cooling time steps between 200 and 500 years. Identical behavior is seen between 200 and 500 years after the refinement. The change in number density of 241Pu during cooling reveals that the dramatic increase in 241Pu uncertainty can be attributed to the drop in its concentration. Because the concentration of 241Pu remains nearly unchanged from 500 to 10,000 years, the uncertainty remains constant. Figures 3.18a and 3.19a depict the uncertainty in minor actinides during cooling. 243Am and 244,245Cm have constant uncertainties which are high at 9%, 9%, and 13%, respectively. The uncertainty of 241Am decreases from 3% at 0.1 year to around 1.5% at 10 years, then stays constant for 1,000 years until rising to 13% in 10,000 years. From 0.1 to 20 years, the uncertainty of 237Np is about 2.5%, then declines to 1% between 200 and 10,000 years. The impact of fission yield uncertainties on the major and minor actinides during cooling is small at below 1%.
Figure 3.20a depicts the fission product uncertainty caused by cross section during cooling.
After discharge, the uncertainty of 134Cs is 4% in the first 100 years, then drops to roughly 2.5%
after that. Because of its long half-life of 2.3 million years, 135Cs uncertainty remains constant at 0.5%. 137Cs uncertainty is 0% till 500 years, then climbs to around 4% at 5,000 years, and then drops to around 2.5% at 10,000 years. 144Ce has a 0% uncertainty until around 20 years, then rises to 8.5% at 50 years, and then drops to 2.5% at 10,000 years. 154Eu has a constant uncertainty below 0.5% until 200 years, after which it rises to almost 3% at 500 years. The rise in 137Cs, 144Ce, and
154Eu uncertainties can be attributed to the drop in their concentrations. In the first 10 years of cooling, 134Cs, 154Eu, and 144Ce (and its decay product) are significant for SNF decay heat. After
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20 years, 137Cs and its decay product are essential to decay heat, as seen in Table 3.4, where the decay product 137mBa of 137Cs is one of the major contributors to decay heat at 23 years. During cooling, the impact of fission yield uncertainties on the fission products remains significant and bigger than the fission yield-related uncertainty of major and minor actinides.
Figure 3.21 depicts the uncertainties of radiation source terms during cooling. Due to the cross section, this uncertainty is within 1% up until 100 years for the activity, decay heat, and gamma source, although this uncertainty is gradually rising. Furthermore, from 100 to 10,000 years, the uncertainties of all three radiation source terms due to cross section are under 1.5%, with the exception of the gamma source uncertainty, which climbs to roughly 4% around 5,000 to 10,000 years. The largest uncertainty caused by cross section occurs at long cooling times for the activity, decay heat, and gamma source. The pattern in the decay heat uncertainty is similar to that seen in previously studied PWR and BWR spent fuel assemblies [57,59]. Furthermore, the trend seen for the decay heat, neutron, and gamma source uncertainty is similar to that described in the reference [58]. Up to 100 years cooling time, fission products are the dominant contributor to the source terms. After 100 years of cooling, the actinides become the main source term contributors, when many fission product would have decayed. The neutron source uncertainty caused by cross section, on the other hand, is substantially higher, rising from 4% at 0.1 year to 9% at roughly 10 years, before falling to between 4% and 6%. Figure 3.21 depicts the maximum cross section induced neutron source uncertainty for short cooling time of roughly 10 years, when the total neutron source is mostly contributed to by 244Cm (96.6% ). In general, the uncertainties of the primary neutron source 244Cm dominate the neutron source uncertainty at less than 100 years.
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(a) (b)
(c)
Figure 3.16. Uncertainties of uranium isotopes during cooling caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.17. Uncertainties of plutonium isotopes during cooling caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.18. Uncertainties of minor actinides during cooling caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.19. Uncertainties of curium isotopes during cooling caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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(a) (b)
(c)
Figure 3.20. Uncertainties of fission products during cooling caused by (a) cross section (b) design parameters and operating conditions and (c) fission yield.
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Figure 3.21. SNF source term uncertainties during cooling caused by cross section, fission yield and modeling parameters.
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