A nuclear data-based uncertainty propagation methodology is extended to enable propagation of manufacturing/technological data (TD) uncertainties in a burn-up calculation problem, taking into account correlation terms between Boltzmann and Bateman terms. The methodology is applied to reactivity and power distributions in a Material Testing Reactor benchmark. Due to the inherent statistical behavior of manufacturing tolerances, Monte Carlo sampling method is used for determining output perturbations on integral quantities. A global sensitivity analysis (GSA) is performed for each manufacturing parameter and allows identifying and ranking the influential parameters whose tolerances need to be better controlled. We show that the overall impact of some TD uncertainties, such as uranium enrichment, or fuel plate thickness, on the reactivity is negligible because the different core areas induce compensating effects on the global quantity. However, local quantities, such as power distributions, are strongly impacted by TD uncertainty propagations. For isotopic concentrations, no clear trends appear on the results.
Sensitivity analysis (SA) methods are invaluable tools allowing the study of how the uncertainty in the model output relies on the different sources of uncertainty in the model inputs [
Tolerance analysis is also becoming an important tool for nuclear engineering design. This seemingly arbitrary task of assigning tolerances can have a large effect on the cost and performance of manufactured products, such as fuel design and fabrication. However, the fact of propagating tolerances instead of uncertainties does not lead to a representative approach of the errors because; in this case, only a bias is taken into account. It is then imperative to understand what kind of physical data creates and propagates uncertainties on the neutronics parameters for both safety and performance reasons. In Material Testing Reactors (MTR), the performance parameters can be core fuel cycle or isotope production. Therefore, it is necessary to calculate isotopic concentrations uncertainties in the reactor core.
We focus in this paper on technological data propagation, with a special attention to uranium enrichment and plate thickness, as example of manufacturing uncertainties propagation. In a general
In general, there are no specific guidelines for allocating tolerances for any component, but [
After a reminder of the theoretical approach and the implementation of tolerance analysis in the MC propagation methodology and UQ in coupled Boltzmann/Bateman problem, a practical example is given for complete depletion calculation, based on a Material Testing Reactor (MTR) benchmark. This latter is described, and the associated tolerance data, based on an actual series of manufacturing feedback, are detailed. One will focus on two main technological parameters: uranium enrichment of the plates as well as their thicknesses.
The uncertainty propagation will be performed for two different integral quantities: the reactivity, that is, a more global parameter, and the power factor (i.e., the plate fission rate distribution), more sensitive to local variations. A particular focus on the concentrations of some important isotopes will also be made.
The method used to evaluate the uncertainties comes from complete work performed in [
The complete evaluation of propagated uncertainties on neutronics parameters requires a precise knowledge of both nuclear data and manufacturing uncertainties. If the primers are relatively well known and characterized through consistent covariance matrices, such as the latest ENDF/B-VII.1 [
The statistical nature of uncertainty analysis naturally relies on the use of Monte Carlo sampling methodology. Monte Carlo sampling methods can be used to perform uncertainty propagation throughout the whole core calculation process. The manufacture of a technological item is simulated, for example, by creating a set of component dimensions with small random changes to simulate natural process variations. In this case, a Gaussian model can be selected as a statistical distribution of uncertainties, and tolerances can be chosen as variances values at
Next, the resulting assembly dimensions are calculated from the simulated set of component dimensions. The numbers of outliers that fall outside the specification limits are then counted. Sample sizes generally range between 5,000 and 100,000, based on the required accuracy of the simulation. The accuracy of Monte Carlo sampling increases with larger sample sizes. Obviously, the computational effort of large sample sizes can be significant, but Monte Carlo sampling offers many advantages because of its flexibility. It also allows the generation of a sample of uncertain inputs. We then obtain a sample corresponding to the outputs of the calculation code.
Of course, the best and more rigorous way is to get actual measurement of each series of manufacturing parameters that would allow building the propagated bias on integral parameters between the theoretical core (i.e., without tolerance) and the actual (i.e.,
The benchmark used is the present paper is a Material Testing Reactor based on 20% enriched 235U
A fuel assembly is made of 22 1.27 mm thick Zircaloy plates (in green). Each plate contains a 0.51 mm thick
Geometric representation of the benchmark.
The benchmark study is performed in 15 energy groups using the APOLLO2.8.3 deterministic code [
For the sake of the present work, two technological parameters and their associated tolerances will be studied. In the following, they will be noted UO2MB for “UO2 mass balance” and PTh for “plate thickness.”
Perturbation at the beginning of the calculation enables assessing global sensitivities.
The statistical distributions are built by calculating standard deviations and average values of manufacturing parameters. We consider, in the following, that measurements of technological parameters are available. An example of this kind of measurement is presented on Figure
Statistical model of manufacturing uncertainty.
As shown in [
The sampling can be performed using a multidimensional Gaussian law whose probability function is given by
R (
The
During the calculation process, it is possible to extract concentrations values at each evolution step.
Using (
We write
As
In this paragraph, we discuss the results obtained on both manufacturing parameters propagated in the calculation code. In the following, the shapes of the outputs follow a normal law. Its variance is here called “uncertainty” and mean is here called “bias.”
Calculations are performed with perturbations of the plate thickness. For each calculation, the thickness is sampled in its distribution law.
The results related to reactivity uncertainty are shown in Figure
Uncertainty on reactivity (pcm) induced by PTh.
However, we observe an important impact on the bias (compared to the standard deviation). This impact is a model bias, coming from the mesh perturbations and from the average of measured values compared to theoretical value (Figure
As for the reactivity, the perturbation of plate thickness has a relatively weak impact on the power factors (Figure
Uncertainty on power factors (%) induced by PTh.
For the UO2MB parameter, the results are slightly different and are detailed in the next 2 paragraphs.
The propagated uncertainty values for reactivity, presented on Figure
Uncertainty on reactivity (pcm) induced by UO2MB.
For the power factors, we observe a random map (Figure
Uncertainty on power factors (%) coming from UO2MB.
The total propagated uncertainty coming from both nuclear data (fission yield and cross sections) and manufacturing data is obtained by applying the process previously described in [
For the reactivity (cf. Figure
Uncertainty on the reactivity during depletion (pcm at
The power factors map uncertainties are principally caused by the UO2MB parameter except at the core-reflector interface. We observe in Figure
Uncertainty on power factors (% at
For the isotopic concentrations, no clear trend can be extracted from the results. The next figures reproduce cumulated propagated uncertainties for several isotopic concentrations. Some present an uncertainty coming principally from nuclear data while others come from manufacturing data. We showed in [
Uncertainty on local isotopic concentrations for some Pu241 and Xe135 at 48000 MW·d/t (% at
Uncertainty on local isotopic concentrations for some Sm149 and Pu239 at 48000 MW·d/t (% at
Uncertainty on local isotopic concentrations for some Gd155 and U235 at 48000 MW·d/t (% at
Table
Summary of isotopic concentrations uncertainties.
Position isotope | Core centre | Core periphery |
---|---|---|
Pu241 | 2,0% | 4,0–4,5% |
Xe135 | 11,5% | 10,0–11,5% |
Sm149 | 3,5% | 1,0–1,5% |
Pu239 | 1,0–1,1% | 1,0–1,1% |
Gd155 | 12,0–15,0% | 8,0–12,0% |
U235 | 0,8–1,2% | 0,8–1,2% |
A methodology for manufacturing uncertainty propagation based on Monte Carlo sampling has been presented and tested on a MTR benchmark. An adequate use of the manufacturer information enables simulating different realisations of the core. Those realisations enable getting estimators of standard deviation for different neutronics parameters such as reactivity, power factors, and local isotopic concentrations. Two manufacturing parameters are propagated in the study. One is the UO2 mass balance in the fuel and the other is a geometric characteristic, the fuel plate thickness.
It is shown that the propagation of these manufacturing parameters uncertainties on reactivity is negligible. This comes from the fact that the perturbed core elements are numerous. So, globally for the core, compensations arise and contribute to reducing the reactivity perturbation, which is a global parameter. As an example, the propagation of a 2,5% uncertainty on the fuel plate thickness, coupled with a 0.2% uncertainty on the UO2 mass balance, induces a combined effect on the reactivity of less than 20 pcm (10−5
For local parameters, the results are different. The mass balance strongly impacts the power factors map during all the depletion cycle. Although the uncertainties decrease during depletion, because of the fuel consumption, it remains important (approximately 1% at
In a second step, a complete uncertainty propagation of all uncertain data (nuclear data such as cross sections and fission yields and manufacturing data) through the depletion cycle has been performed to observe the combination of all these uncertainties. It is shown that even if these manufacturing data uncertainties can be neglected for reactivity, there is no clear trend for isotopic concentrations due to the coupling of different sources of uncertainties. For example, contributions to the 135Xe uncertainty during depletion come from both nuclear data and manufacturing data. From [
The authors declare that they have no competing interests.