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All issues Volume 4 (2018) EPJ Nuclear Sci. Technol., 4 (2018) 42 Full HTML
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EPJ Nuclear Sci. Technol.
Volume 4, 2018
Special Issue on 4th International Workshop on Nuclear Data Covariances, October 2–6, 2017, Aix en Provence, France – CW2017
Article Number 42
Number of page(s) 7
Section Applied Covariances
DOI https://doi.org/10.1051/epjn/2018023
Published online 14 November 2018

© H. Iwamoto et al., published by EDP Sciences, 2018

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

To promote research and development of nuclear technology for various applications such as accelerator-driven systems, the Generation-IV reactors, and production of medical radioisotopes, the Belgian Nuclear Research Centre (SCK∙CEN) has proposed a cutting-edge research reactor combined with a proton accelerator, MYRRHA [1,2]. From the viewpoint of ensuring safety margins and reducing uncertainty in the MYRRHA design parameters, uncertainty quantification of reactor physics parameters is one of the most important tasks. To this end, nuclear data sensitivity and uncertainty (S/U) analyses have been extensively conducted for various MYRRHA core configurations using different calculation tools, geometric models, and nuclear data libraries [3–5]; these works have focused on the effective neutron multiplication factor keff as the primary neutronic safety parameter. The effective delayed neutron fraction βeff can be ranked second in the list of neutronic safety parameters, because, besides of reactor kinetics, it is used to determine other design and safety parameters such as control rod worth and Doppler coefficient.

Continuous-energy Monte Carlo transport codes such as the Monte Carlo N-Particle transport code MCNP [6] have been widely used in calculating not only keff but also its nuclear data sensitivities and kinetic parameters including βeff. Although these codes have no capability to directly calculate the βeff sensitivities owing to technical cumbersomeness, it is approximately expressed as a function of two different keff sensitivities by the so-called “k-ratio method [7]”; this indicates that uncertainty in βeff can be quantified by the sensitivity method from the approximate βeff sensitivities and evaluated covariance data of the nuclear data library. Although this method has been applied to the βeff S/U analysis with a deterministic code SUSD3D for MYRRHA in the studies of Kodeli [8], within the seventh framework programme solving CHAllenges in Nuclear DAta (CHANDA) project [9], the analysis using the Monte Carlo transport code has not yet been tackled.

The k-ratio method itself is currently subdivided into two techniques: the prompt k-ratio method [7] and the modified k-ratio method proposed by Chiba [10,11]. Our previous study [12] by MCNP for a critical configuration of the VENUS-F zero-power reactor at the SCK∙CEN site [13] demonstrated that the prompt k-ratio method involves large statistical uncertainty in the calculated βeff sensitivities, and it would be currently difficult to reduce it only by increasing the number of neutron source histories. On the other hand, we also demonstrated that Chiba’s modified k-ratio method can alleviate this kind of problem.

In this study, we conducted the S/U analysis of βeff for two types of MYRRHA configurations (i.e. critical mode and subcritical mode) using Chiba’s modified k-ratio method. βeff and its sensitivities were calculated using the MCNP version 6.1.1, and JENDL-4.0u [14,15] was used as the nuclear data library since it contains covariance data of delayed neutron yield .

2 MYRRHA core models

MYRRHA is designed to operate both in critical mode as a lead-bismuth cooled fast reactor and in subcritical mode driven by 600-MeV linear proton accelerator. Figures 1 and 2 show horizontal sectional views of the MYRRHA critical and subcritical configurations, respectively, which are homogenized on assembly level. The analyses were carried out for the critical and subcritical core configurations at beginning of cycle using the assembly-based homogenized models [16]. As illustrated, the critical and subcritical cores consist of 78 and 58 fuel assemblies (FAs), respectively; these are loaded with MOX with high Pu content. Besides FAs, the cores contains in-pile sections for material testing, irradiation rigs for medical isotope production, and subassemblies containing safety and control rods. More details of the MYRRHA core design are given in Van den Eynde et al. [2].

thumbnail Fig. 1

Horizontal sectional view of MYRRHA critical core.
thumbnail Fig. 2

Horizontal sectional view of MYRRHA subcritical core.

3 Methodology

3.1 The βeff sensitivities

Chiba’s modified k-ratio method describes βeff as (1) where a(≠ 0) is a scaling factor; k is the effective multiplication factor calculated using the nuclear data library JENDL-4.0u; is the effective multiplication factor calculated using the library in which is multiplied with (a + 1) (Appendix A). If a = − 1, then Chiba’s modified k-ratio method reduces to the prompt k-ratio method. The statistical uncertainty in βeff propagated from δk and is expressed as follows: (2)

Here correlations were disregarded for the sake of simplicity. Using the definition of the sensitivity and equation (1), the βeff sensitivity to parameter x is expressed as follows: (3) (4) where and are the sensitivities of k and to the parameter x, respectively. The statistical uncertainty in  is expressed as (5)where and are the statistical uncertainties in and , respectively.

3.2 The βeff uncertainties

Using the sensitivity profile obtained by the above-mentioned methods, the βeff uncertainty (standard deviation) due to nuclear data was evaluated by the uncertainty propagation law considering covariance, which is expressed as (6) where cov(z, g ; z′, g′) denotes a (g, g′ ; z, z′) component of variance–covariance matrix (g, g′: energy group, z, z′: reaction), which was obtained by processing covariance data stored in JENDL-4.0u with NJOY [17] and ERRORJ [18]. In this analysis, we employed eight parameters: fission, neutron capture, elastic scattering, inelastic scattering, and (n, 2n) reaction cross sections for major constituent materials: U, Pu, 241Am, 16O, 56Fe, Pb, and 209Bi, as well as prompt and delayed neutron yields and prompt neutron spectra for U and Pu isotopes. Here correlations between the reactions were also considered. Absence of covariance data for delayed neutron spectrum in JENDL-4.0u did not allow us to estimate the contribution to the βeff and confirm the conclusion made by Kodeli [19] on its significance.

4 Results and discussion

4.1 Effective delayed neutron fraction

Figure 3 shows comparisons of the calculated βeff with different scaling factors (a = − 1, 1, 5, 10, 15, 20, and 25) for critical and subcritical cores. For each case, the keff value was calculated with 2.5 × 108 histories (2.5 × 105 source histories per cycle times 103 cycles) in the MCNP calculation flow. Here the value calculated directly by the adjoint weighting method [20] for the same number of histories for critical and subcritical cores were estimated to be 323 ± 4 and 320 ± 4 pcm, respectively. It can be seen that the calculated statistical uncertainty decreases with increase in a, which was about |a| times smaller than that by the prompt k ratio method (a = − 1). In addition, we see that the calculated βeff values exceeding their 1σ statistical uncertainties decrease as a increases. The similar trend can be seen in the previous study conducted for the VENUS-F reactor using MCNP [12] and the benchmark analysis for fast neutron systems conducted by Chiba using the deterministic transport code CBG [10,21]; this reason is linked to the approximation used in equation (1).

thumbnail Fig. 3

Comparison of the βeff values for different scaling factors calculated using Chiba’s modified k-ratio method and that derived using the adjoint weighting method. The error bars and the band with pale blue color indicate 1σ statistical uncertainty.

4.2 Sensitivity

Figures 4 and 5 show comparisons of the βeff sensitivity profiles of 239Pu fission and elastic scattering cross sections and 239Pu and for the critical core with different scaling factors, respectively. As demonstrated in Iwamoto et al. [12], the statistical uncertainty at a = 1 is very large for all parameters except ; this is caused by the small difference between and (Appendix A). In contrast, the statistical uncertainty for is negligibly small for all the selected a values; this is owing to an approximation of (Appendix A). In addition, as with the βeff values, decrease as a increases. However, it should be noted that the βeff sensitivities change within about 1σ statistical uncertainties, while the nominal values of βeff tend to exceed their 1σ statistical uncertainties. This indicates that the influence of the change in the βeff sensitivities that results from increasing a on the uncertainty quantification of βeff is expected to be small.

Figures 6 and 7 show the βeff sensitivity profiles with respect to 239Pu and 238U reaction parameters for the critical core with a = 20, respectively, in which the statistical uncertainties appear to be sufficiently small. Table 1 summarizes the major βeff sensitivities together with 1σ statistical uncertainties which were calculated with a = 20. As expected from the definition of βeff, of major fuel materials (i.e. 239Pu and 238U) shows positive high sensitivities; in contrast, demonstrates negative sensitivities.

thumbnail Fig. 4

Comparisons of βeff sensitivity profile of 239Pu fission (left) and elastic scattering (right) cross sections for the critical core with different scaling factors. The pale blue color around the blue line indicates 1σ statistical uncertainty.
thumbnail Fig. 5

Comparisons of βeff sensitivity profile of 239Pu (left) and (right) for the critical core with different scaling factors. The pale blue color around the blue line indicates 1σ statistical uncertainty.
thumbnail Fig. 6

239Pu sensitivity profile for the critical core (a = 20).
thumbnail Fig. 7

238U sensitivity profile for the critical core (a = 20).
Table 1

Major βeff sensitivities with 1σ statistical uncertainities (a = 20, top 15).

4.3 Uncertainty

Table 2 lists the βeff uncertainties due to nuclear data with different scaling factors for the critical and subcritical cores. Small scaling factors produce large both uncertainty values and their statistical uncertainties. This arises from the large sensitivities with large statistical uncertainties in the sensitivity profile as mentioned above; these statistical uncertainties can be reduced by increasing a. In addition, we see from Table 2 that, as  a increases, the calculated total βeff uncertainties for the critical and subcritical cores approach values of 2.2% and 2.0%, respectively.

Table 3 summarizes top 15 contributors to the βeff uncertainty together with 1σ statistical uncertainties. Overall, the statistical uncertainties are small enough to identify the main contributors; namely, it can be concluded that the βeff uncertainty is dominated by , followed by the elastic and inelastic scattering cross sections, for both cores.

Table 2

Calculated βeff uncertainty with different scaling factors.

Table 3

Major contributors to the βeff uncertainty with 1σ statistical uncertainties (a = 20, top 15).

5 Conclusion

We have conducted the nuclear data S/U analysis of βeff for critical and subcritical cores of the MYRRHA reactor using the MCNP code. The βeff sensitivities were calculated by Chiba’s modified k-ratio method. Although the nominal βeff values appear to worsen as a increases, comparing the βeff sensitivities and their statistical uncertainties calculated with different scaling factors shows that the βeff sensitivities are more stable to the a change than the nominal βeff values, and that a value of a = 20 is the most suitable for the uncertainty quantification. Using the calculated βeff sensitivities and the JENDL-4.0u covariance data, the βeff uncertainties for the critical and subcritical cores have been determined to be 2.2 ± 0.2% and 2.0 ± 0.2%, respectively, which are dominated by of 239Pu and 238U.

To account for the optimal a value more clearly, further investigation, especially for non-linear effects on and , is needed. Moreover, it would be of interest to compare our results with those previously performed using the deterministic code within the ongoing CHANDA project [8].

Appendix A Chiba’s modified k-ratio method

In equation (1), the effective neutron multiplication factor keff is obtained by solving the following neutron transport equation: (A.1) where  is the angular neutron flux at position r with direction Ω and energy E; , , and  are the transport, scattering, and fission operators, respectively, which are expressed as follows: (A.2) (A.4)

Here , , and  represent the macroscopic total, scattering, and fission cross sections, respectively; χ and are total fission neutron spectrum and fission neutron yield, respectively. The effective multiplication factor in equation (1) is obtained by solving a fictitious neutron transport equation below: (A.5) where is the fictitious angular neutron flux obtained from equation (A.5); is a fission operator defined as follows: (A.6)

Using the relationship between prompt, delayed, and total neutrons: (A.7)

Equation (A.6) can be rewritten as (A.8)

It is noted that a corresponds to the fractional change of and that, if a = − 1 is adopted, only prompt neutrons are taken into account; in this case, Chiba’s modified k-ratio method reduces to the prompt k-ratio method [22]. In the MCNP calculation, is obtained using a nuclear data library in which is multiplied with (a + 1).

According to the adjoint-based perturbation theory, the sensitivity of k to parameter x is expressed as [23] (A.9) where the bracket 〈⁡ , ⁡ 〉 represents integration over the phase space;  is the adjoint function of ψ; λ = 1/k; Σx is the macroscopic cross section corresponding to parameter x;  is the scattering operator for parameter x;  is the fission operator for parameter x.

As an example of parameters except for , let us consider neutron capture cross section σcap. Since σcap involves neither scattering nor fission, the sensitivity of k to σcap is written by (A.10)

Further, the sensitivity of to σcap is written by (A.11) where and Σcap represents the macroscopic capture cross section. We see from equation (A.6) that, if a approaches zero, becomes ℱ; and hence , , and become ψ, , and k, respectively. Namely, if a small value is chosen for a, the difference between equations (A.10) and (A.11) is also small.

Similarly, the sensitivities of k and k to  are written by (A.12) and (A.13)respectively. As with the case of other parameters, as a approaches zero, the limit of equation (A.13) of a equals equation (A.12). If a particular value is chosen for a to the extent that the approximations of , , and  are applicable, equation (A.13) can be expressed as follows: (A.14)

Thus, if a = 1 is used, then is about twice as large as ; namely, it follows that there is a large difference between the two.

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Cite this article as: Hiroki Iwamoto, Alexey Stakovskiy, Luca Fiorito, Gert Van den Eynde, Sensitivity and uncertainty analysis of βeff for MYRRHA using a Monte Carlo technique, EPJ Nuclear Sci. Technol. 4, 42 (2018)

All Tables

Table 1

Major βeff sensitivities with 1σ statistical uncertainities (a = 20, top 15).

In the text
Table 2

Calculated βeff uncertainty with different scaling factors.

In the text
Table 3

Major contributors to the βeff uncertainty with 1σ statistical uncertainties (a = 20, top 15).

In the text

All Figures

thumbnail Fig. 1

Horizontal sectional view of MYRRHA critical core.
In the text
thumbnail Fig. 2

Horizontal sectional view of MYRRHA subcritical core.
In the text
thumbnail Fig. 3

Comparison of the βeff values for different scaling factors calculated using Chiba’s modified k-ratio method and that derived using the adjoint weighting method. The error bars and the band with pale blue color indicate 1σ statistical uncertainty.
In the text
thumbnail Fig. 4

Comparisons of βeff sensitivity profile of 239Pu fission (left) and elastic scattering (right) cross sections for the critical core with different scaling factors. The pale blue color around the blue line indicates 1σ statistical uncertainty.
In the text
thumbnail Fig. 5

Comparisons of βeff sensitivity profile of 239Pu (left) and (right) for the critical core with different scaling factors. The pale blue color around the blue line indicates 1σ statistical uncertainty.
In the text
thumbnail Fig. 6

239Pu sensitivity profile for the critical core (a = 20).
In the text
thumbnail Fig. 7

238U sensitivity profile for the critical core (a = 20).
In the text

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