- Email: [email protected]
The Role of Uncertainty Quantification for Reactor Physics
Available online at www.sciencedirect.com
Nuclear Data Sheets 123 (2015) 68–73 www.elsevier.com/locate/nds
The Role of Uncertainty Quantification for Reactor Physics M. Salvatores,1, 2, ∗ G. Aliberti,3 and G. Palmiotti2 1 Consultant Idaho National Laboratory, Idaho Falls, ID 83415, USA 3 Argonne National Laboratory, Argonne, IL 60439, USA (Received 12 May 2014; revised received 5 August 2014; accepted 8 August 2014) 2
The quantification of uncertainties is a crucial step in design. The comparison of a-priori uncertainties with the target accuracies, allows to define needs and priorities for uncertainty reduction. In view of their impact, the uncertainty analysis requires a reliability assessment of the uncertainty data used. The choice of the appropriate approach and the consistency of different approaches are discussed. I.
INTRODUCTION
The role of uncertainty quantification has been stressed and has been the object of several assessments in the past (see e.g. Refs. [1] and [2] among many others), in particular in relation to design requirements for safety assessments, design margins definition and optimization, both for the reactor core and for the associated fuel cycles. The use of integral experiments has been advocated since many years, and recently re-assessed [3] in order to reduce uncertainties and to define new reduced “aposteriori” uncertainties. While uncertainty quantification in the case of existing power plants benefits from a large data base of operating reactor experimental results, innovative reactor systems (reactor and associated fuel cycles) should rely on limited power reactor experiment data bases and on a number of past integral experiments that should be shown to be representative enough. Moreover, in some cases, in particular related to innovative fuel cycle performance and feasibility assessment, nuclear data uncertainties are the only available information. Uncertainty quantification in that case becomes a tool for detecting potential show stoppers associated to specific fuel cycle strategies, besides the challenges related to fuel properties, fuel processing chemistry and material performance issues.
II.
THE DESIGNER DILEMMA
The quantification of uncertainties is a crucial step in different phases of a nuclear system design. In a preliminary (conceptual) design phase, the comparison of calcu-
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.012 0090-3752/© 2014 Elsevier Inc. All rights reserved.
lation scheme (nuclear data and modeling) a-priori uncertainties with the target accuracies for the most important design parameters, allows to define needs and priorities for calculation scheme improvement and uncertainty reduction. The designer analysis establishes the quantified penalties due to uncertainties beyond the target accuracy range and their impact on the design (e.g. extra margins on fuel performances, choice of alternative or back-up solutions etc.). Successively, and in parallel with preliminary design, the choice of the most adapted approach to uncertainty reduction could be done according to timeframe, project schedule etc., but also according to safety requirements (e.g. demonstration of validated uncertainties). In view of their impact, the uncertainty analysis requires a reliability assessment of the uncertainty data that have been used. The choice of the appropriate approach can be a dilemma for the designer. In practically all case, the uncertainty quantification for design will imply the use of experiments (past experiments or ad-hoc experiments still to be performed): • Performance of a full series of design oriented experiments (critical mass, reaction rate distributions, reactivity coefficients, control rod worth etc.) in a representative reactor mock-up. This is the most ambitious (in terms of resources deployment), but not necessarily the most effective or even feasible approach (facility availability, cost, difficulty to achieve representativity etc.). If available, the uncertainty reduction by integral parameter R is a function of the a-priori covariance data D [4] 2 ΔR02 = ΔR02 · 1 − rRE , (1) + SR DSE + , rRE = + (2) SR DSR SE DSE where the SE and the SR are the sensitivity vectors
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
of the experiments and of the design parameters, respectively.
III.
• A more flexible approach is to use a large set of “representative” integral experiments and to perform a global assimilation or adjustment that allows to obtain an “adjusted” nuclear data set and an “a-posteriori” covariance matrix D . This new covariance matrix can be used to assess the new (reduced) uncertainty for each integral design parameter R of interest [4] + ΔR02 = SR D SR .
M. Salvatores et al.
INTEGRAL PARAMETERS AND THEIR SEPARATED COMPONENTS
A. The Na Void Coefficient in an Axial Heterogeneous Innovative Fast Reactor Core
Low void reactivity coefficient is an innovative feature of the French ASTRID design [5]. Studies performed on a representative configuration [6] confirm a global, full core and upper structure void reactivity coefficient that is fairly small and even negative. However, the sodium void reactivity is the result of compensation of large components with different sign, according to the formulation, in diffusion theory, given below
(3)
Δk 1 + c = {NN a σN a,j Φj Φj k F j + −NN a σN a (j → k) Φj Φ+ k − Φj
Both approaches 1 and 2 rely a) on the reliability and completeness of the covariance data; b) on the reliability of the integral experimental uncertainties; c) on the capability to detect possible systematic errors in the experiments and their overall consistency and d) on the drastic reduction of modeling errors.
−δDj
j,k
ΔΦj ΔΦ+ j dV −
i
Ni
j
a δσi,j Φj Φ+ j }
= ANa − SNa − L − AselfSh . (4)
• A third approach can be (and has been) envisaged that relies on the existence or performance of selected integral experiments that provide information on “elemental” phenomena or on separated individual physics effects. This approach can provide practical “uncertainties”, derived from the calculation/experiment analysis, the observed C/E dispersion and consistent with the experimental uncertainties, on each “elemental” phenomena and/or on separated physics effects. It could alternatively also provide “bias factors” extracted from the residual C/E s that can be combined appropriately.
For instance in the case of the reactivity change associated to voiding the sodium in the fissile + internal fertile + upper plenum and calculated in R-Z S4 P1 the total value of -1024 pcm is the result of the difference between -3587 pcm (leakage component, L), and +2563 pcm (nonleakage component, of which +2222 pcm of spectral component, ANa − SNa and +341 pcm of self-shielding variation, ASelfSh ). In effect, in order to be more conservative, one should calculate the uncertainties by component and combine them with some degree of correlations, with the completely uncorrelated hypothesis being the most conservative. In order to evaluate the uncertainty per component the previously mentioned case of the fissile + internal fertile + upper plenum region sodium void was considered. This is the situation that has been considered the most relevant to further safety calculations. Table I shows the uncertainty results (in pcm) for the total (all components) sodium void reactivity, obtained using the COMMARA-2 covariance data [7]. The breakdown by isotope and reaction is provided. From the total value, considering a 95% confidence interval (2σ), an uncertainty of 1$ [8] should be associated to this reactivity coefficient for the successive safety calculations. Tables II and III show the uncertainties for, respectively, the leakage and non-leakage components. In terms of contributions one can notice that for the non-leakage component the uncertainty is largely dominated by the anisotropic elastic cross sections of sodium. Therefore, one has to be very careful with the related covariance evaluation. The non-leakage uncertainty is also larger, by itself, than that of the total effect. In combining the non-leakage and leakage component uncertainties the safest assumption is a conservative assumption of simple sum of the uncertainties. With that kind of assumption, and, again
This last method (that relies on criteria b), c) and d) indicated for methods 1 and 2) is of particular interest when applied to design parameters that result from the compensation of several separated (or elemental) effects, potentially of different sign and potentially of comparable magnitude. This is the case of most reactivity coefficients and of the core reactivity evolution with time. An interesting feature of this approach is represented by the possibility to compare the overall uncertainties obtained both using the a-priori covariance data (method 2) and the “uncertainty” derived from the analysis of a selected set of integral experiments (method 3). Consistency of the two approaches will be a strong argument in support of the robustness of the a-priori covariance data. In the following paragraph we will discuss two typical cases: the sodium void reactivity coefficient in an innovative FR and the general case of the reactivity loss/cycle. In both cases it is possible to express the integral parameter as a sum of physical components that can in principle be measured separately. In the examples we will indicate how old experiments can be used or how some simple experiments can be designed to meet the needs of this approach. 69
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
a 95% confidence interval, a total uncertainty of 2$ is obtained. The presence of negative values in Table II and III is due either to negative correlations or different signs of sensitivity coefficients for the correlated cross sections. This effect appears only for uncertainty components, while the total uncertainty, as it should be expected, is always positive and should be interpreted as a standard deviation.
TABLE III. Uncertainty results (in pcm) for the non-leakage component of the sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope 241 Am 16 O 52 Cr 56 Fe 23 Na 58 Ni 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 238 U Total
TABLE I. Uncertainty results (in pcm) for the total (all components) sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope 241 Am 16 O 52 Cr 56 Fe 23 Na 58 Ni 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 238 U Total
σcap σf iss 1 0 1 0 1 0 14 0 2 0 1 0 3 8 9 21 9 10 5 10 5 3 22 4 30 27
ν 0 0 0 0 0 0 3 7 16 2 3 13 22
σel σinel 0 0 2 1 17 2 43 13 25 47 4 2 0 1 3 7 2 4 0 1 1 0 25 110 59 121
χ 0 0 0 0 0 0 4 11 12 0 0 0 16
M. Salvatores et al.
P 1el Sum 0 1 0 2 0 17 12 49 65 84 0 4 0 9 0 27 0 24 0 12 0 7 0 116 66 158
σcap σf iss 1 0 0 0 1 0 16 0 -2 0 1 0 4 8 9 21 12 5 5 11 7 1 34 1 42 26
ν 0 0 0 0 0 0 1 7 4 2 1 1 9
σel σinel 0 0 12 0 1 1 24 11 88 33 1 1 0 1 -2 10 -1 4 0 3 0 1 -11 57 91 68
χ P 1el Sum 0 0 1 0 0 12 0 0 2 0 4 31 0 144 172 0 0 1 1 0 9 5 0 26 4 0 15 0 0 13 0 0 7 0 0 66 7 145 191
often using the hypothesis of independent uncertainties. In the case of axial heterogeneous cores, Ref. [9] reports results obtained in the Zero Power Plutonium Reactor (ZPPR) facility. Some typical results obtained at the time are shown in Table IV taken from Ref. [9]. Step one TABLE IV. Sodium void results for inner core zones in ZPPR17A. Step (mm) a 152 to 330 330 to 508 0 to 152 507 to 787
TABLE II. Uncertainty results (in pcm) for the leakage component of the sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope σcap σf iss 241 Am 0 0 16 O 1 0 52 Cr 1 0 56 Fe 3 0 23 Na 3 0 58 Ni 0 0 238 Pu 1 5 239 Pu 8 12 240 Pu 4 10 241 Pu 2 3 242 Pu 2 4 238 U 19 5 18 Total 22
ν 0 0 0 0 0 0 3 4 16 1 3 15 22
σel σinel 0 0 13 0 20 1 57 5 15 62 6 1 0 0 -2 5 1 2 0 3 1 0 23 69 68 93
χ 0 0 0 0 0 0 3 15 13 0 0 0 20
P 1el Sum 0 0 0 13 0 20 13 59 64 90 0 6 0 7 0 22 0 23 0 5 0 5 0 77 65 139
a b
¢/kgb Nonleakage Leakage Total C-E 0.314 0.3462 -0.0546 0.2916 -0.022 0.086 0.2744 -0.1750 0.0995 +0.013 0.474 0.4965 -0.0599 0.4366 -0.037 -0.147 0.1432 -0.1572 -0.0140 +0.133
Voiding symmetrically in each half. 1 ¢= 0.01 $, see Ref. [8].
and three represent non-leakage dominated experiments, while in step two but in particular in step four the leakage becomes comparable or higher than the non-leakage component. Figure 1 shows in more detail the sequence of the axial void experiments. The non-leakage component shows a regular behavior as expected, since it is, at least in a first approximation, proportional to the square of the flux. The leakage component, at first approximation proportional to the square of the flux gradient, shows two minima as a result of inward leakage to the axial blanket with a zero component a little below the axial center of the fuel. The experimental uncertainty, as evaluated in [9] can be quantified as shown in Table V. From the observed C-E values, one could deduce variable average discrepancy values, to be compared to the uncertainties derived from a-priori covariance data. The behavior shown in figure, underline the interest of experiments able to point out specific physical features and that provide average discrepancy values with uncertainties (or alternatively, bias factors together with uncertainties) to be associated to these features. A re-analysis of this and other similar experiments, should provide the basis for an
As far as the third approach indicated in the previous paragraph, past experiments have been performed in order to provide information on each separate component of the reactivity coefficient (e.g. leakage and non-leakage or “spectral” components), and to derive uncertainties from the observed dispersion of C/E values or, as an alternative, derive bias factors for each separate component together with its appropriate uncertainty. The uncertainty of the total effect was successively derived from the separate component uncertainties or, alternatively, applying the appropriate bias factors to each component of the design calculation and combining the uncertainties, 70
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE VI. Fast reactor system characteristics. Characteristic Power (MWth) Type of fuel TRU/HM Pu/HM MA/HM Coolant Cycle Length (d)
EFR 3560 Oxide 0.236 0.226 0.010 Na 1700
SFR ADMAB (ADS) 900 390 Metal Nitride (U-free) 0.589 1.0 0.533 0.375 0.056 0.625 Na Pb-Bi 155 366
core region index) Δρ(cycle) =
ΔNi (νσf − σa ) i νΣ f i,n i,n n i ΔNF P σF P − . i,n νΣf i,n
FIG. 1. Calculated and measured sodium void worth as function of axial position in ZPPR-17A (from Ref. [9]).
The fission product component is represented by a standard “lumped fission product” description. One can rewrite that expression as ΔnK ρK , (6) Δρ(cycle) =
TABLE V. Measured sodium void worth and associated uncertainties in ZPPR-17A. Step (mm) Measured ( ¢/kg) Uncertainty 152 to 330 0.314 ± 0.010 0.086 ± 0.012 330 to 508 0.474 ± 0.010 0 to 152 -0.147 ± 0.025 507 to787
K
where we have assumed that n=1 and that the K index includes both the “i” heavy isotopes and the lumped fission product and K ΔnK = nK F − n0
(7)
(F index indicates the end and the 0 index the beginning of irradiation, respectively). In this case one in principle should use measurements of the reactivity of individual heavy isotopes and of fission products (e.g. by means of sample reactivity measurements performed in several representative neutron spectrum environments) and use the C/E values dispersion and associated uncertainties, possibly to be combined statistically. In the case of the reactivity loss during irradiation, this integral quantity can be measured e.g. in a reactor (e.g. via reactivity compensation with control rods). However, one can measure separately
indirect but independent global validation of the a-priori nuclear data uncertainties mentioned above.
B.
(5)
Reactivity Loss/Cycle
For innovative fuel cycles, the extension of burn-ups for very long, once-through fuel cycles in fast neutron systems or the multiple recycle of TRU with different MA/Pu ratios, also in fast neutron reactors, are approaches intended to better exploit resources and to minimize the production or to drastically reduce the stocks of radioactive wastes and their impact on a geological repository. In all cases, the reactivity balance will change significantly during irradiation and should be monitored carefully, i.e. with an accurate uncertainty quantification. To make a check on the credibility of the uncertainty, the approach “by elemental effect” has been applied to the case of the reactivity loss per cycle in three different systems, characterized by very different fuel types and compositions and by different cycle length (see Table VI). The reactivity variation per cycle can be expressed (in a simple one-group fundamental mode approximation) as the sum of individual isotope i contributions (n being the
• the number density variations of each separated isotope (fissile, fertile, fission products) that contribute to the reactivity variation and • the “reactivity/atom” of the same isotopes. This has been done e.g. at the MASURCA reactor (BALZAC program, see e.g. Ref. [10]) using experiments where, starting from a reference critical configuration, one did replace it (e.g. at the center of the zero-power reactor) with a new composition that differed from the reference by a known amount of one isotope. The measurement of the new reactivity with respect to the critical reference, gives a direct information on the reactivity of that specific isotope. We have performed a test case supposing that such 71
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
“gedanken” experiments did exist for the different isotope reactivity values and we did associate “plausible” uncertainties to each component, derived from the analysis of a large number of experiments (e.g. in different spectra), and the observed dispersion of the C/E as given in Table VII.
M. Salvatores et al.
components are significantly different, as a consequence of the different fuel compositions. The 239 Pu component is in all case predominant, but in the ADMAB case the other Pu isotope components (and in particular 238 Pu) and MA components, not only Am isotopes but Cm isotopes too, play a very significant role. As far as uncertainty, the analysis is performed using sensitivity coefficients Sjcycle associated to the variation of any σj , given by
TABLE VII. Assumed specific reactivity uncertainties by isotope derived from C/E dispersion in different experiments. Isotope Uncertainty 239 Pu ±5% 238 U, 238 Pu, 240 Pu, 241 Pu, 242 Pu ±10% ±20% Am isotopes, Cm isotopes ±15% Lumped FP
σj Δρcycle Δρcycle ∂σj
∂nK K ∂ρK · ρK + Δn . ∂σj ∂σj Sjcycle =
=
σj Δρcycle
K
Using the formulations of Ref. [11] we obtain tF ρK Sjcycle = n∗ σj ndt Δρcycle t0
1 ∗ 1 + Φ p , σj Φ p − Φ∗ , σj Φ , Ifp If
TABLE VIII. Reactivity variation during irradiation for the systems of Table VI. Decomposition by elemental contributions (values in pcm). EFR SFR ADMAB U 2384.2 59.4 0.0 237 Np 9.4 14.1 -23.3 238 Pu -56.3 -122.1 5486.3 239 Pu -7139.3 -2687.4 -9294.3 240 Pu -165.7 -231.9 126.8 241 Pu 1554.7 -463.5 -3467.6 242 Pu -16.1 -40.8 -28.3 241 Am 208.1 27.6 461.0 242m Am 10.6 -13.8 2045.7 243 Am -2.9 9.3 311.5 242 Cm 7.1 6.4 168.4 244 Cm 20.5 14.4 639.8 245 Cm 21.8 -55.7 1036.7 -6775 -670 -1990 FP -10070 -4120 -4510 Total Δρ(cycle) HI uncertainty ±400 (±573) ±150 (±111) ±950 (±812) component FP uncertainty ±1000 ±100 ±300 component Total uncertainty ±1100 ±180 ±1000 238
(8)
K
(9)
where the index “p” refers to the core state at t = tF . Table VIII shows the uncertainties deduced using the assumed reactivity component uncertainties of Table VII. Using the COMMARA-2 [7] covariance data, uncertainty values have also been obtained for the HI component, and are shown in parenthesis. If the uncertainties of Table VII (mostly based on hypothetical or existing experiments) would be confirmed for the cases where the HI component is dominant, as the case of SFR or, at a lesser extent in the case of ADMAB, the remarkable agreement between the uncertainty based on a-priori (in this case COMMARA-2) cross section covariance data and the uncertainty based on the analysis of the elemental components of the reactivity variation, would be a very powerful indirect but independent test of the a-priori covariance data and of the validity of the uncertainties in a wide range of different systems. Finally, it should be kept in mind that specifically tailored experiments, already quoted e.g. in [13], could allow not only to validate separated isotope reactivity effects, but also to separate at an even more fundamental level, absorption and scattering effects.
The assumed values are rather standard, as deduced from the analysis of past experiments [12, 13]. Of course, these values are used here to illustrate an approach and should be carefully re-evaluated (using updated data and methods) before application to a real design case. Table VIII shows the Δρ(cycle) for the three systems defined above, corresponding to their respective cycle length, and their decomposition into elemental components i.e. by isotope and having used a standard lumped fission product description of FP build-up. In the three cases the role of heavy isotopes (HI) and of FP is rather different: for the case of the European Fast Reactor (EFR) with a long irradiation cycle, the Δρ(cycle) is due for ∼70% to FP, while in the Sodium Fast Reactor (SFR) case, the FP component represents only ∼15% of the total effect. The Accelerator Driven Minor Actinide Burner (ADMAB) case is somewhat in between. However, the HI
IV.
CONCLUSIONS
Uncertainty quantification is a major requirement for any innovative reactor system design. Existing covariance data provide a very useful starting point. However, improvement and reduction of uncertainties are needed to meet design target accuracies. Experiments will still play the key role both in the validation of current uncertainties and in the search of reduced uncertainties. However, global and costly experiments of the mock-up type seem today out of reach of most projects. In this respect, a new approach based as far as possible on the experimental validation of separated physics effects 72
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
could play a very significant role.
M. Salvatores et al.
der DOE Idaho operations office contract DE-AC0705ID14517 and under Argonne National Laboratory contract DE-AC02-06CH11357.
Acknowledgements: Work was supported by the U.S. Department of Energy, Office of Nuclear Energy, un-
[1] M. Salvatores, G. Palmiotti, “LMFBR design parameter uncertainties and target accuracies,” Ann. Nucl. Energy 12, 291 (1985). [2] M. Salvatores et al., “Needs and issues of covariance data application,” Nucl. Data Sheets 109, 2725-2732 (2008). [3] M. Salvatores et al., “Methods and Issues for the Combined Use of Integral Experiments and Covariance Data,” Report NEA/NSC/WPEC/DOC(2013)445 (OECD/NEA, December 2013). [4] G. Palmiotti, M. Salvatores, “The Role of Experiments and of Sensitivity Analysis in Simulation Validation Strategies with Emphasis on Reactor Physics,” Ann. Nucl. Energy 52, 10-21 (2013). [5] F. Varaine et al., “Pre-Conceptual Design Study of ASTRID Core,” Proceedings of ICAPP 12, Chicago, USA (June 24-28, 2012). [6] J. Bess et al., “DOE-CEA Benchmark on SFR ASTRID Innovative Core: Neutronic and Safety Transients Simulation,” Proc. Int. Conf. FR13, Paris (March 2013). [7] M.W. Herman et al., “COMMARA-2.0 Neutron Cross Section Covariance Library,” Report BNL-94830-2011, Brookhaven National Laboratory (2011).
[8] Reactivity coefficient is expressed in “dollars”, $, defined as ratio of the reactivity ρ, in units of δk/k, to the effective delayed neutron fraction, ρ/β. In this scale 1 ¢ = 0.01$. [9] S.B. Brumback, P.J. Collins, “Experiments and Analysis for an Axially Heterogeneous Liquid-Metal Reactor Assembly at the Zero-Power Physics Reactor,” Nucl. Sci. Eng. 103, 219 (1989). [10] R. Soule et al., “The Experimental Balzac Program at MASURCA in Support of the design of SuperPhenix 2,” Proc. of the Topical Meeting on Reactor Physics and Safety, Saratoga Springs, NY, USA (September 17 -19, 1986). [11] J.M. Kallfelz, G. Bruna, G. Palmiotti, and M. Salvatores, “Burn-up calculations with time-dependent generalized perturbation theory,” Nucl. Sci. Eng. 62, 304 (1977). [12] K.S. Smith and R.W. Schaefer, “Recent Developments in the Small Sample Reactivity Discrepancy,” Nucl. Sci. Eng. 87, 314 (1984). [13] K. Dietze, “Integral Test of JENDL-3.2 Data by Reanalysis of Sample Reactivity Measurements at Fast Critical Facilities,” Report JNC TN9400 2001-043 (2001).
73
Nuclear Data Sheets 123 (2015) 68–73 www.elsevier.com/locate/nds
The Role of Uncertainty Quantification for Reactor Physics M. Salvatores,1, 2, ∗ G. Aliberti,3 and G. Palmiotti2 1 Consultant Idaho National Laboratory, Idaho Falls, ID 83415, USA 3 Argonne National Laboratory, Argonne, IL 60439, USA (Received 12 May 2014; revised received 5 August 2014; accepted 8 August 2014) 2
The quantification of uncertainties is a crucial step in design. The comparison of a-priori uncertainties with the target accuracies, allows to define needs and priorities for uncertainty reduction. In view of their impact, the uncertainty analysis requires a reliability assessment of the uncertainty data used. The choice of the appropriate approach and the consistency of different approaches are discussed. I.
INTRODUCTION
The role of uncertainty quantification has been stressed and has been the object of several assessments in the past (see e.g. Refs. [1] and [2] among many others), in particular in relation to design requirements for safety assessments, design margins definition and optimization, both for the reactor core and for the associated fuel cycles. The use of integral experiments has been advocated since many years, and recently re-assessed [3] in order to reduce uncertainties and to define new reduced “aposteriori” uncertainties. While uncertainty quantification in the case of existing power plants benefits from a large data base of operating reactor experimental results, innovative reactor systems (reactor and associated fuel cycles) should rely on limited power reactor experiment data bases and on a number of past integral experiments that should be shown to be representative enough. Moreover, in some cases, in particular related to innovative fuel cycle performance and feasibility assessment, nuclear data uncertainties are the only available information. Uncertainty quantification in that case becomes a tool for detecting potential show stoppers associated to specific fuel cycle strategies, besides the challenges related to fuel properties, fuel processing chemistry and material performance issues.
II.
THE DESIGNER DILEMMA
The quantification of uncertainties is a crucial step in different phases of a nuclear system design. In a preliminary (conceptual) design phase, the comparison of calcu-
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.012 0090-3752/© 2014 Elsevier Inc. All rights reserved.
lation scheme (nuclear data and modeling) a-priori uncertainties with the target accuracies for the most important design parameters, allows to define needs and priorities for calculation scheme improvement and uncertainty reduction. The designer analysis establishes the quantified penalties due to uncertainties beyond the target accuracy range and their impact on the design (e.g. extra margins on fuel performances, choice of alternative or back-up solutions etc.). Successively, and in parallel with preliminary design, the choice of the most adapted approach to uncertainty reduction could be done according to timeframe, project schedule etc., but also according to safety requirements (e.g. demonstration of validated uncertainties). In view of their impact, the uncertainty analysis requires a reliability assessment of the uncertainty data that have been used. The choice of the appropriate approach can be a dilemma for the designer. In practically all case, the uncertainty quantification for design will imply the use of experiments (past experiments or ad-hoc experiments still to be performed): • Performance of a full series of design oriented experiments (critical mass, reaction rate distributions, reactivity coefficients, control rod worth etc.) in a representative reactor mock-up. This is the most ambitious (in terms of resources deployment), but not necessarily the most effective or even feasible approach (facility availability, cost, difficulty to achieve representativity etc.). If available, the uncertainty reduction by integral parameter R is a function of the a-priori covariance data D [4] 2 ΔR02 = ΔR02 · 1 − rRE , (1) + SR DSE + , rRE = + (2) SR DSR SE DSE where the SE and the SR are the sensitivity vectors
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
of the experiments and of the design parameters, respectively.
III.
• A more flexible approach is to use a large set of “representative” integral experiments and to perform a global assimilation or adjustment that allows to obtain an “adjusted” nuclear data set and an “a-posteriori” covariance matrix D . This new covariance matrix can be used to assess the new (reduced) uncertainty for each integral design parameter R of interest [4] + ΔR02 = SR D SR .
M. Salvatores et al.
INTEGRAL PARAMETERS AND THEIR SEPARATED COMPONENTS
A. The Na Void Coefficient in an Axial Heterogeneous Innovative Fast Reactor Core
Low void reactivity coefficient is an innovative feature of the French ASTRID design [5]. Studies performed on a representative configuration [6] confirm a global, full core and upper structure void reactivity coefficient that is fairly small and even negative. However, the sodium void reactivity is the result of compensation of large components with different sign, according to the formulation, in diffusion theory, given below
(3)
Δk 1 + c = {NN a σN a,j Φj Φj k F j + −NN a σN a (j → k) Φj Φ+ k − Φj
Both approaches 1 and 2 rely a) on the reliability and completeness of the covariance data; b) on the reliability of the integral experimental uncertainties; c) on the capability to detect possible systematic errors in the experiments and their overall consistency and d) on the drastic reduction of modeling errors.
−δDj
j,k
ΔΦj ΔΦ+ j dV −
i
Ni
j
a δσi,j Φj Φ+ j }
= ANa − SNa − L − AselfSh . (4)
• A third approach can be (and has been) envisaged that relies on the existence or performance of selected integral experiments that provide information on “elemental” phenomena or on separated individual physics effects. This approach can provide practical “uncertainties”, derived from the calculation/experiment analysis, the observed C/E dispersion and consistent with the experimental uncertainties, on each “elemental” phenomena and/or on separated physics effects. It could alternatively also provide “bias factors” extracted from the residual C/E s that can be combined appropriately.
For instance in the case of the reactivity change associated to voiding the sodium in the fissile + internal fertile + upper plenum and calculated in R-Z S4 P1 the total value of -1024 pcm is the result of the difference between -3587 pcm (leakage component, L), and +2563 pcm (nonleakage component, of which +2222 pcm of spectral component, ANa − SNa and +341 pcm of self-shielding variation, ASelfSh ). In effect, in order to be more conservative, one should calculate the uncertainties by component and combine them with some degree of correlations, with the completely uncorrelated hypothesis being the most conservative. In order to evaluate the uncertainty per component the previously mentioned case of the fissile + internal fertile + upper plenum region sodium void was considered. This is the situation that has been considered the most relevant to further safety calculations. Table I shows the uncertainty results (in pcm) for the total (all components) sodium void reactivity, obtained using the COMMARA-2 covariance data [7]. The breakdown by isotope and reaction is provided. From the total value, considering a 95% confidence interval (2σ), an uncertainty of 1$ [8] should be associated to this reactivity coefficient for the successive safety calculations. Tables II and III show the uncertainties for, respectively, the leakage and non-leakage components. In terms of contributions one can notice that for the non-leakage component the uncertainty is largely dominated by the anisotropic elastic cross sections of sodium. Therefore, one has to be very careful with the related covariance evaluation. The non-leakage uncertainty is also larger, by itself, than that of the total effect. In combining the non-leakage and leakage component uncertainties the safest assumption is a conservative assumption of simple sum of the uncertainties. With that kind of assumption, and, again
This last method (that relies on criteria b), c) and d) indicated for methods 1 and 2) is of particular interest when applied to design parameters that result from the compensation of several separated (or elemental) effects, potentially of different sign and potentially of comparable magnitude. This is the case of most reactivity coefficients and of the core reactivity evolution with time. An interesting feature of this approach is represented by the possibility to compare the overall uncertainties obtained both using the a-priori covariance data (method 2) and the “uncertainty” derived from the analysis of a selected set of integral experiments (method 3). Consistency of the two approaches will be a strong argument in support of the robustness of the a-priori covariance data. In the following paragraph we will discuss two typical cases: the sodium void reactivity coefficient in an innovative FR and the general case of the reactivity loss/cycle. In both cases it is possible to express the integral parameter as a sum of physical components that can in principle be measured separately. In the examples we will indicate how old experiments can be used or how some simple experiments can be designed to meet the needs of this approach. 69
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
a 95% confidence interval, a total uncertainty of 2$ is obtained. The presence of negative values in Table II and III is due either to negative correlations or different signs of sensitivity coefficients for the correlated cross sections. This effect appears only for uncertainty components, while the total uncertainty, as it should be expected, is always positive and should be interpreted as a standard deviation.
TABLE III. Uncertainty results (in pcm) for the non-leakage component of the sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope 241 Am 16 O 52 Cr 56 Fe 23 Na 58 Ni 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 238 U Total
TABLE I. Uncertainty results (in pcm) for the total (all components) sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope 241 Am 16 O 52 Cr 56 Fe 23 Na 58 Ni 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 238 U Total
σcap σf iss 1 0 1 0 1 0 14 0 2 0 1 0 3 8 9 21 9 10 5 10 5 3 22 4 30 27
ν 0 0 0 0 0 0 3 7 16 2 3 13 22
σel σinel 0 0 2 1 17 2 43 13 25 47 4 2 0 1 3 7 2 4 0 1 1 0 25 110 59 121
χ 0 0 0 0 0 0 4 11 12 0 0 0 16
M. Salvatores et al.
P 1el Sum 0 1 0 2 0 17 12 49 65 84 0 4 0 9 0 27 0 24 0 12 0 7 0 116 66 158
σcap σf iss 1 0 0 0 1 0 16 0 -2 0 1 0 4 8 9 21 12 5 5 11 7 1 34 1 42 26
ν 0 0 0 0 0 0 1 7 4 2 1 1 9
σel σinel 0 0 12 0 1 1 24 11 88 33 1 1 0 1 -2 10 -1 4 0 3 0 1 -11 57 91 68
χ P 1el Sum 0 0 1 0 0 12 0 0 2 0 4 31 0 144 172 0 0 1 1 0 9 5 0 26 4 0 15 0 0 13 0 0 7 0 0 66 7 145 191
often using the hypothesis of independent uncertainties. In the case of axial heterogeneous cores, Ref. [9] reports results obtained in the Zero Power Plutonium Reactor (ZPPR) facility. Some typical results obtained at the time are shown in Table IV taken from Ref. [9]. Step one TABLE IV. Sodium void results for inner core zones in ZPPR17A. Step (mm) a 152 to 330 330 to 508 0 to 152 507 to 787
TABLE II. Uncertainty results (in pcm) for the leakage component of the sodium void reactivity of the fissile + internal fertile + upper plenum region. Isotope σcap σf iss 241 Am 0 0 16 O 1 0 52 Cr 1 0 56 Fe 3 0 23 Na 3 0 58 Ni 0 0 238 Pu 1 5 239 Pu 8 12 240 Pu 4 10 241 Pu 2 3 242 Pu 2 4 238 U 19 5 18 Total 22
ν 0 0 0 0 0 0 3 4 16 1 3 15 22
σel σinel 0 0 13 0 20 1 57 5 15 62 6 1 0 0 -2 5 1 2 0 3 1 0 23 69 68 93
χ 0 0 0 0 0 0 3 15 13 0 0 0 20
P 1el Sum 0 0 0 13 0 20 13 59 64 90 0 6 0 7 0 22 0 23 0 5 0 5 0 77 65 139
a b
¢/kgb Nonleakage Leakage Total C-E 0.314 0.3462 -0.0546 0.2916 -0.022 0.086 0.2744 -0.1750 0.0995 +0.013 0.474 0.4965 -0.0599 0.4366 -0.037 -0.147 0.1432 -0.1572 -0.0140 +0.133
Voiding symmetrically in each half. 1 ¢= 0.01 $, see Ref. [8].
and three represent non-leakage dominated experiments, while in step two but in particular in step four the leakage becomes comparable or higher than the non-leakage component. Figure 1 shows in more detail the sequence of the axial void experiments. The non-leakage component shows a regular behavior as expected, since it is, at least in a first approximation, proportional to the square of the flux. The leakage component, at first approximation proportional to the square of the flux gradient, shows two minima as a result of inward leakage to the axial blanket with a zero component a little below the axial center of the fuel. The experimental uncertainty, as evaluated in [9] can be quantified as shown in Table V. From the observed C-E values, one could deduce variable average discrepancy values, to be compared to the uncertainties derived from a-priori covariance data. The behavior shown in figure, underline the interest of experiments able to point out specific physical features and that provide average discrepancy values with uncertainties (or alternatively, bias factors together with uncertainties) to be associated to these features. A re-analysis of this and other similar experiments, should provide the basis for an
As far as the third approach indicated in the previous paragraph, past experiments have been performed in order to provide information on each separate component of the reactivity coefficient (e.g. leakage and non-leakage or “spectral” components), and to derive uncertainties from the observed dispersion of C/E values or, as an alternative, derive bias factors for each separate component together with its appropriate uncertainty. The uncertainty of the total effect was successively derived from the separate component uncertainties or, alternatively, applying the appropriate bias factors to each component of the design calculation and combining the uncertainties, 70
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE VI. Fast reactor system characteristics. Characteristic Power (MWth) Type of fuel TRU/HM Pu/HM MA/HM Coolant Cycle Length (d)
EFR 3560 Oxide 0.236 0.226 0.010 Na 1700
SFR ADMAB (ADS) 900 390 Metal Nitride (U-free) 0.589 1.0 0.533 0.375 0.056 0.625 Na Pb-Bi 155 366
core region index) Δρ(cycle) =
ΔNi (νσf − σa ) i νΣ f i,n i,n n i ΔNF P σF P − . i,n νΣf i,n
FIG. 1. Calculated and measured sodium void worth as function of axial position in ZPPR-17A (from Ref. [9]).
The fission product component is represented by a standard “lumped fission product” description. One can rewrite that expression as ΔnK ρK , (6) Δρ(cycle) =
TABLE V. Measured sodium void worth and associated uncertainties in ZPPR-17A. Step (mm) Measured ( ¢/kg) Uncertainty 152 to 330 0.314 ± 0.010 0.086 ± 0.012 330 to 508 0.474 ± 0.010 0 to 152 -0.147 ± 0.025 507 to787
K
where we have assumed that n=1 and that the K index includes both the “i” heavy isotopes and the lumped fission product and K ΔnK = nK F − n0
(7)
(F index indicates the end and the 0 index the beginning of irradiation, respectively). In this case one in principle should use measurements of the reactivity of individual heavy isotopes and of fission products (e.g. by means of sample reactivity measurements performed in several representative neutron spectrum environments) and use the C/E values dispersion and associated uncertainties, possibly to be combined statistically. In the case of the reactivity loss during irradiation, this integral quantity can be measured e.g. in a reactor (e.g. via reactivity compensation with control rods). However, one can measure separately
indirect but independent global validation of the a-priori nuclear data uncertainties mentioned above.
B.
(5)
Reactivity Loss/Cycle
For innovative fuel cycles, the extension of burn-ups for very long, once-through fuel cycles in fast neutron systems or the multiple recycle of TRU with different MA/Pu ratios, also in fast neutron reactors, are approaches intended to better exploit resources and to minimize the production or to drastically reduce the stocks of radioactive wastes and their impact on a geological repository. In all cases, the reactivity balance will change significantly during irradiation and should be monitored carefully, i.e. with an accurate uncertainty quantification. To make a check on the credibility of the uncertainty, the approach “by elemental effect” has been applied to the case of the reactivity loss per cycle in three different systems, characterized by very different fuel types and compositions and by different cycle length (see Table VI). The reactivity variation per cycle can be expressed (in a simple one-group fundamental mode approximation) as the sum of individual isotope i contributions (n being the
• the number density variations of each separated isotope (fissile, fertile, fission products) that contribute to the reactivity variation and • the “reactivity/atom” of the same isotopes. This has been done e.g. at the MASURCA reactor (BALZAC program, see e.g. Ref. [10]) using experiments where, starting from a reference critical configuration, one did replace it (e.g. at the center of the zero-power reactor) with a new composition that differed from the reference by a known amount of one isotope. The measurement of the new reactivity with respect to the critical reference, gives a direct information on the reactivity of that specific isotope. We have performed a test case supposing that such 71
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
“gedanken” experiments did exist for the different isotope reactivity values and we did associate “plausible” uncertainties to each component, derived from the analysis of a large number of experiments (e.g. in different spectra), and the observed dispersion of the C/E as given in Table VII.
M. Salvatores et al.
components are significantly different, as a consequence of the different fuel compositions. The 239 Pu component is in all case predominant, but in the ADMAB case the other Pu isotope components (and in particular 238 Pu) and MA components, not only Am isotopes but Cm isotopes too, play a very significant role. As far as uncertainty, the analysis is performed using sensitivity coefficients Sjcycle associated to the variation of any σj , given by
TABLE VII. Assumed specific reactivity uncertainties by isotope derived from C/E dispersion in different experiments. Isotope Uncertainty 239 Pu ±5% 238 U, 238 Pu, 240 Pu, 241 Pu, 242 Pu ±10% ±20% Am isotopes, Cm isotopes ±15% Lumped FP
σj Δρcycle Δρcycle ∂σj
∂nK K ∂ρK · ρK + Δn . ∂σj ∂σj Sjcycle =
=
σj Δρcycle
K
Using the formulations of Ref. [11] we obtain tF ρK Sjcycle = n∗ σj ndt Δρcycle t0
1 ∗ 1 + Φ p , σj Φ p − Φ∗ , σj Φ , Ifp If
TABLE VIII. Reactivity variation during irradiation for the systems of Table VI. Decomposition by elemental contributions (values in pcm). EFR SFR ADMAB U 2384.2 59.4 0.0 237 Np 9.4 14.1 -23.3 238 Pu -56.3 -122.1 5486.3 239 Pu -7139.3 -2687.4 -9294.3 240 Pu -165.7 -231.9 126.8 241 Pu 1554.7 -463.5 -3467.6 242 Pu -16.1 -40.8 -28.3 241 Am 208.1 27.6 461.0 242m Am 10.6 -13.8 2045.7 243 Am -2.9 9.3 311.5 242 Cm 7.1 6.4 168.4 244 Cm 20.5 14.4 639.8 245 Cm 21.8 -55.7 1036.7 -6775 -670 -1990 FP -10070 -4120 -4510 Total Δρ(cycle) HI uncertainty ±400 (±573) ±150 (±111) ±950 (±812) component FP uncertainty ±1000 ±100 ±300 component Total uncertainty ±1100 ±180 ±1000 238
(8)
K
(9)
where the index “p” refers to the core state at t = tF . Table VIII shows the uncertainties deduced using the assumed reactivity component uncertainties of Table VII. Using the COMMARA-2 [7] covariance data, uncertainty values have also been obtained for the HI component, and are shown in parenthesis. If the uncertainties of Table VII (mostly based on hypothetical or existing experiments) would be confirmed for the cases where the HI component is dominant, as the case of SFR or, at a lesser extent in the case of ADMAB, the remarkable agreement between the uncertainty based on a-priori (in this case COMMARA-2) cross section covariance data and the uncertainty based on the analysis of the elemental components of the reactivity variation, would be a very powerful indirect but independent test of the a-priori covariance data and of the validity of the uncertainties in a wide range of different systems. Finally, it should be kept in mind that specifically tailored experiments, already quoted e.g. in [13], could allow not only to validate separated isotope reactivity effects, but also to separate at an even more fundamental level, absorption and scattering effects.
The assumed values are rather standard, as deduced from the analysis of past experiments [12, 13]. Of course, these values are used here to illustrate an approach and should be carefully re-evaluated (using updated data and methods) before application to a real design case. Table VIII shows the Δρ(cycle) for the three systems defined above, corresponding to their respective cycle length, and their decomposition into elemental components i.e. by isotope and having used a standard lumped fission product description of FP build-up. In the three cases the role of heavy isotopes (HI) and of FP is rather different: for the case of the European Fast Reactor (EFR) with a long irradiation cycle, the Δρ(cycle) is due for ∼70% to FP, while in the Sodium Fast Reactor (SFR) case, the FP component represents only ∼15% of the total effect. The Accelerator Driven Minor Actinide Burner (ADMAB) case is somewhat in between. However, the HI
IV.
CONCLUSIONS
Uncertainty quantification is a major requirement for any innovative reactor system design. Existing covariance data provide a very useful starting point. However, improvement and reduction of uncertainties are needed to meet design target accuracies. Experiments will still play the key role both in the validation of current uncertainties and in the search of reduced uncertainties. However, global and costly experiments of the mock-up type seem today out of reach of most projects. In this respect, a new approach based as far as possible on the experimental validation of separated physics effects 72
The Role of Uncertainty . . .
NUCLEAR DATA SHEETS
could play a very significant role.
M. Salvatores et al.
der DOE Idaho operations office contract DE-AC0705ID14517 and under Argonne National Laboratory contract DE-AC02-06CH11357.
Acknowledgements: Work was supported by the U.S. Department of Energy, Office of Nuclear Energy, un-
[1] M. Salvatores, G. Palmiotti, “LMFBR design parameter uncertainties and target accuracies,” Ann. Nucl. Energy 12, 291 (1985). [2] M. Salvatores et al., “Needs and issues of covariance data application,” Nucl. Data Sheets 109, 2725-2732 (2008). [3] M. Salvatores et al., “Methods and Issues for the Combined Use of Integral Experiments and Covariance Data,” Report NEA/NSC/WPEC/DOC(2013)445 (OECD/NEA, December 2013). [4] G. Palmiotti, M. Salvatores, “The Role of Experiments and of Sensitivity Analysis in Simulation Validation Strategies with Emphasis on Reactor Physics,” Ann. Nucl. Energy 52, 10-21 (2013). [5] F. Varaine et al., “Pre-Conceptual Design Study of ASTRID Core,” Proceedings of ICAPP 12, Chicago, USA (June 24-28, 2012). [6] J. Bess et al., “DOE-CEA Benchmark on SFR ASTRID Innovative Core: Neutronic and Safety Transients Simulation,” Proc. Int. Conf. FR13, Paris (March 2013). [7] M.W. Herman et al., “COMMARA-2.0 Neutron Cross Section Covariance Library,” Report BNL-94830-2011, Brookhaven National Laboratory (2011).
[8] Reactivity coefficient is expressed in “dollars”, $, defined as ratio of the reactivity ρ, in units of δk/k, to the effective delayed neutron fraction, ρ/β. In this scale 1 ¢ = 0.01$. [9] S.B. Brumback, P.J. Collins, “Experiments and Analysis for an Axially Heterogeneous Liquid-Metal Reactor Assembly at the Zero-Power Physics Reactor,” Nucl. Sci. Eng. 103, 219 (1989). [10] R. Soule et al., “The Experimental Balzac Program at MASURCA in Support of the design of SuperPhenix 2,” Proc. of the Topical Meeting on Reactor Physics and Safety, Saratoga Springs, NY, USA (September 17 -19, 1986). [11] J.M. Kallfelz, G. Bruna, G. Palmiotti, and M. Salvatores, “Burn-up calculations with time-dependent generalized perturbation theory,” Nucl. Sci. Eng. 62, 304 (1977). [12] K.S. Smith and R.W. Schaefer, “Recent Developments in the Small Sample Reactivity Discrepancy,” Nucl. Sci. Eng. 87, 314 (1984). [13] K. Dietze, “Integral Test of JENDL-3.2 Data by Reanalysis of Sample Reactivity Measurements at Fast Critical Facilities,” Report JNC TN9400 2001-043 (2001).
73