- Email: [email protected]
Needs and Issues of Covariance Data Application
Nuclear Data Sheets 109 (2008) 2725–2732 www.elsevier.com/locate/nds
Needs and Issues of Covariance Data Application M. Salvatores,1,2,3∗ G. Palmiotti,1 G. Aliberti,2 H.Hiruta,1 R. McKnight,2 P. Obloˇzinsk´ y,4 W.S. Yang2 1
Idaho National Laboratory, NSE Division, 2525 Fremont Ave. P.O. Box 1625, Idaho Falls, ID 83415, USA 2 Argonne National Laboratory, NE Division, Argonne, IL 60439, USA 3 CEA-Cadarache, 13108 St-Paul-Lez-Durance, France and 4 National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA (Received July 28, 2008) Needs and issues related to covariance data application are provided and several examples are given to show their impact on important design parameters that affects both the economic and the safety of advanced nuclear reactors.
I.
DESIGN TARGET UNCERTAINTIES
Recently, extensive sensitivity and uncertainty studies [1, 2] and the availability of new neutron cross section covariance data [3] have allowed the preliminary quantification of the impact of current nuclear data uncertainties on the design parameters of the major Gen-IV systems. This, in particular, holds for sodium-cooled fast reactors (SFRs) with different fuels (oxide or metal), fuel composition (e.g., different Pu/TRU ratios) and different conversion ratios† . These studies have pointed out that present uncertainties on the nuclear data should be significantly reduced, in order to obtain the full benefit from advances in modeling and simulation. Only a parallel effort in advanced simulation and in nuclear data improvement will be able to provide designers with more general and well validated calculation tools to meet design target accuracies. Current as well as targeted uncertainties for some of the most important SFR design parameters have been assessed [4], applicable to the Advanced Burner Reactor (ABR) design. A typical example, related to core neutronics, is given in Table I. This table gives, for each parameter, the respective contribution to the current estimated uncertainties of both input data and simulation tools. It is also important to realize that covariance data can have a significant impact on innovative design features, as it will be illustrated in the next Section. Then, in subsequent two Sections we will first discuss uncertainties and target accuracies, followed by explanation of the combined used of covariance data and integral experiments.
∗ Electronic
address: [email protected] [†] Gen-IV stands for new generation of nuclear power reactors. TRU means transuranium materials.
0090-3752/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.nds.2008.11.001
II.
PHYSICS ISSUES AND IMPACT OF UNCERTAINTIES
A wide range of systems has been investigated, both within the Advanced Fuel Cycle Initiative (AFCI) and the Global Nuclear Energy Partnership (GNEP) initiatives, even if no final design choices have been made yet. Some expected new significant features (for the core and fuel cycle) depend heavily on nuclear data knowledge and uncertainties. Moreover, design margins, optimization and choice of options will depend on an accepted accuracy level in simulations. This is an important issue, since economy and safety are at stake. Typical examples of nuclear data dependent innovative design features are: • Cores with low reactivity loss during the cycle, • Cores with increased inventory of Minor Actinides (MA) in the fuel, • Cores with no uranium blankets. Both core design and the associated fuel cycle features have to be considered.
A.
Optimization of Na void reactivity versus Internal Breeding Gain
In order to optimize the irradiation time, any neutronically efficient design will have a low reactivity loss during the cycle Δρ, i.e., an Internal Breeding Gain, IBG, close to zero Δρ ≈ {(νσf − σa )U 238 − (νσf − σa )P u239 } × IBG cycle −(νσf − σa )U 238 + σcF P ,
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE I: Current and targeted uncertainties for some Sodium-cooled Fast Reactors (SFR) design parameters. Neutronics: Core Parameter Current uncertainty Targeted Parameter Current uncertainty Targeted Uncertainty Uncertainty Input Data Origin Modeling Input Data Origin Modeling (A Priori) Origin (A Priori) Origin Multiplication Fac1.5% 0.5% 0.3% Reactivity Co7% 15% 7% tor, (Δkeff /keff ) efficients: Total Power 1% 3% 2% Reactivity 20% 20% 10% Peak Coefficients: Component Power 1% 6% 3% Fast Flux for 7% 3% 3% Distribution Damage Conversion Ratio 5% 2% 2% Kinetics 10% 5% 5% (Absolute Value) Parameters Control Rod Worth: 5% 6% 5% Local Nuclide 5% 3% 2% Element Densities: Major Actinides Control Rod Worth: 5% 4% 2% Local Nu30% 10% 10% Total clide Densities: Minor Actinides Burnup Reactivity 0.7% 0.5% 0.3% Fuel Decay Heat 10% 3% 5% Swing (Δkeff /keff ) at Shutdown 6 5 4
Eta
where ν is the number of neutrons per fission, σf is the fission cross section and σa is the absorption cross section. A reactivity loss close to zero allows for the designing of very long life cores, with evident economic benefits. Moreover, such design requires an excess reactivity at the beginning of cycle near zero, avoiding potentially large reactivity insertions (control rod run out, handling error, etc.), which can represent a potential safety issue. However, it has been demonstrated in the past [5] that, in the case of Na-cooled Fast reactors, there is a very tight correlation between the reactivity loss per cycle and the reactivity associated with the coolant voiding of the core. It is also well known that in the case of the Na-cooled Fast Reactors, it is important to assess the sign and the magnitude of the reactivity coefficient associated to a coolant voiding of the core. This reactivity coefficient can be expressed, e.g. using diffusion perturbation theory, as follows
Cm245 3 2 1 0 1.E+0
U238
Pu239
Am241 Pu238 1.E+1
Cm244 1.E+2
1.E+3
1.E+4
Np237 1.E+5
1.E+6
1.E+7
1.E+8
Energy [eV]
FIG. 1: η = ν σf /σa energy shape for selected actinides.
related to the change in self shielding of e.g. 238 U, due to the Na density variation and in general ASelfSh is positive and small. As indicated above, the sign of the spectral component 1 Δkeff c {NNa = σNa,j Φj Φ+ SNa is determined by the energy shape of the adjoint flux, j keff F j which, at first approximation, is related to the energy + + dependence of η = νσf /σa . − NNa σNa (j → k)Φj (Φk − Φj ) Since different fissile isotopes (239 Pu, 235 U, 233 U) and j,k Minor Actinides have significantly different η shapes (see + a Fig. 1), the result is very different adjoint flux shapes, } δDj ∇Φj ∇Φ+ dV − N δσ Φ Φ − i i,j j j j according to the actual TRU composition of the core, in j i j particular at high energy, and, as a consequence, different = ANa − SNa − L − ASelfSh. values of the (positive) Na void scattering component. As for the spectral component SNa one can have SNa >0 The trade-off between the Na void coefficient and the + or <0 according to the sign of (Φ+ Φ ). The leakage Δρ/cycle plays an important role in core design (e.g., to j k component L is such that L>0 and its absolute value will define number and position of control rods in the core). increase with the decrease of the geometrical dimensions. In the case of a core with a low Δρ/cycle, the spectral Finally, the absorption component ANa is positive and, component becomes dominant, according to the qualitain general, relatively small. The ASelfSh component is tive scheme indicated in Fig. 2. 2726
Needs and Issues... ĭ+
NUCLEAR DATA SHEETS
Adjoint flux behavior behavior as Adjoint functionflux of E: as fonction of E:
) ~
20
M. Salvatores et al.
Variation %
Variation (%)
A
Increase of positive Na void reactivity coefficient
10 A
Q6 f 6 a DB 2
0 B
- 10
D C D
- 20
High U-238/Pu-239 ratio
Low U-238/Pu-239 ratio
A : Vidange sodium B : Doppler C : Burn-up D : Bêta effectf
C
- 30 B
- 40 Act. Mineurs
Chargés % MA in (%) fuel
- 50
Positive Na void reactivity coeff.
Decrease of reactivity loss/cycle
0
Negative Na void reactivity coeff.
2
Decrease of ȕeff
FIG. 2: Schematic adjoint flux energy shape and Na void reactivity/IBG relation.
The presence of a high content of Minor Actinides in future innovative fast reactors is related to the specific mission assigned to fast reactors in order to minimize the amount of waste produced by a power fleet. The content of MA in the fuel varies according to specific strategies. For example, fast neutron cores that should stabilize the MA inventory in the fuel cycle within a TRU homogeneous recycle strategy, will have a MA content of 2-5% and definitely less than 10% (see e.g. Ref. [6]). A “burner” fast reactor can have MA content in the fuel which varies according to the conversion ratio and the type of fuel. For example, fuel with no uranium could accommodate up to 50% MA. However, the physics characteristics of most MA isotopes (e.g., low or very low fraction of delayed neutrons, threshold fission, etc.), will have an impact on key reactivity coefficients, as shown in figure 3, related to an SFR. The features shown in that figure indicate that there can be limitations in the amount of MA that can be added safely to the fuel, because of the need to keep an acceptable compromise between the deterioration of some reactivity coefficients (e.g., Na-void and Doppler coefficients) and the improvement e.g., of the reactivity loss/cycle. Once more, all these feature result from a very delicate balance of effects of different sign, and data uncertainties can play a very significant role, both in a design opti-
6
8
10
Max MA content ~ 2.5% for large size SFR (EFR) and 5-10 % for medium/small size (Phenix)
Decrease of Doppler reactivity coefficient
FIG. 3: Variation of SFR parameters with minor actinide (MA) content in the fuel.
As a consequence, there can be a significant potential impact of nuclear data uncertainties on core feasibility and on its safety assessment. For example, the good compromise on IBG value and Na-void reactivity based on nominal values of both parameters could be revised to take into account uncertainties with practical consequences on, e.g., the core burn-up: an IBG more negative should be found in order to have a less positive Na-void coefficient, with a higher reactivity loss/cycle and consequently a potential reduction of the burn up.
B. Fast reactor cores with high content of Minor Actinides: impact on reactivity coefficients (case of a SFR)
4
TABLE II: keff sensitivity (%) of 56 Fe data. Experiment/ Type Capture Elastic Inelastic Configuration ZPR3-53 Blanket -0.11 0.76 -0.02 ZPR3-54 Reflector -1.40 16.5 1.50 CIRANO Reflector -1.50 6.28 -0.24 ZPPR-15 Blanket -1.54 1.67 -2.55 ABR-Metal Reflector -1.49 3.05 -3.06 ABR Oxide Reflector -1.73 1.79 -3.29
Total 0.63 16.6 4.55 2.43 -1.51 -3.23
mization phase and in more advanced design phases, e.g. during the safety case assessment. For example, if one wants to limit a) the increase of the Na-void reactivity, b) the decrease of the Doppler effect and c) the decrease of the delayed effective neutron fraction, one has to limit the amount of MA in the fuel, e.g. to 3-5%; however, nuclear data uncertainties have to be accounted for, and they can suggest a more conservative choice of the MA content. This impact can be more significant for “burner” cores with potentially a higher MA content in the fuel.
C.
Blanket versus reflector in fast reactors
Nonproliferation concerns suggest avoiding uranium blankets in the next generation FRs, and replacing them with reflectors. However, the presence of reflectors induces neutron spectrum transients at the core/reflector interface that need specific calculation methods to be described appropriately [7]. It has been shown that the spectrum transient effects are also sensitive to data issues [7]. For example, the keff sensitivity to the different Fe-56 data, in configurations with a blanket or with a reflector is indicated in Table II. Very different sensitivities are shown, as expected. Also in this case, nuclear data uncertainties will have to be taken into account, both in keff and in the power distribution close to the interface or on the peak-to-average
2727
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE III: Isotopic contributions to the decay heat (MeV/s/g) at fuel unloading (case of metal fuel and Pu/MA=0.5 in the feed) for two values of Conversion Ratio. CR=0.0 CR=1.0 Total decay heat 3.08 E+12 3.53 E+10 Pu-238 component 3.00 E+11 6.50 E+9 Am-241 component 5.23 E+10 1.62 E+9 Cm-242 component 6.64 E+11 1.35 E+10 Cm-244 component 2.03 E+12 1.03E+10 %Cm-244 in the fuel 8.63 E-02 5.2E-04
R= 186 cm R= 115 cm R= 73 cm R= 59 cm
S 1 /(1 O1 / O2 ) Probable range for ABR cores
FIG. 4: Spatial sensitivity profile of a central control rod in different fast reactors.
power ratio in the core, with potential consequences on the core optimization (e.g., worth and position of external control rods).
D.
topes, with different decay constants, give comparable contributions. Their build-up during irradiation should be known rather accurately. This means there is a need for accurate capture and fission data in the related Bateman equations, in order to avoid uncertainties greater than about 10% on isotope densities, as indicated in Table I.
III. UNCERTAINTIES AND TARGET ACCURACIES: LESSONS LEARNED WITH WPEC SUBGROUP 26
Control rod spatial sensitivity to nuclear data uncertainties
It is well known that space dependent effects (e.g. the system response to a localized perturbation) are dependent on core size or, more precisely, on the Boltzmann equation eigenvalue separation [8]. Due to this very fundamental feature, the control rod worth in a fast reactor (whatever its coolant) is affected differently by cross section uncertainties according to the size of the core, control rod location and environment. In Fig. 4, the spatial dependence of the sensitivity of a control rod worth is shown in cores of different sizes, each characterized by the separation of the first two eigenvalues of the related Boltzmann operator. In the probable range of future ABR cores, the sensitivity of control rod worth (both for isolated rods but even more for a control rod system in interaction) to data uncertainty can be rather significant and should be accounted for in the core safety assessment.
E.
Some fuel cycle issues
As for the Fuel Cycle, in an assessment of the most important features for the Fuel Cycle, it is important to have a clear understanding (i.e., with low uncertainty) of isotope contributions to specific effects. This drives the related design requirement of Table I. For example, it can be shown [9] that the decay heat at fuel unloading differs by a factor of about 100 for fast reactor cores with a Conversion Ratio CR=0 or =1. The relative isotopic contributions at fuel unloading are shown in Table III for a typical ABR related fuel cycle. In the case with CR=0, Cm-244 is practically the only contributor, while, in the case with CR=1, several iso-
Recent work to assess uncertainties on a wide range of integral parameters and on a wide range of systems, has been performed within an international initiative and a final report is being issued [10]. The studies, that have been performed on a wide range of e.g. fast reactors with different coolants, fuel type or Pu/MA ratios, allow an analysis in more details of some of the effects previously discussed. In particular, uncertainties have been derived for keff , Na-void reactivity, Δρ/cycle, nuclide densities at the end of irradiation etc. For most of the physics phenomena indicated above, the level of uncertainty is high enough to jeopardize optimization and to require introducing costly safety margins. If preliminary design calculations are certainly possible with current nuclear data, the type of uncertainties that have been pointed out will certainly have an important impact on more advanced design phases. In particular, several target accuracies as indicated in Table I will not be met. The studies, performed in the OECD/NEA Working Party on Evaluation Cooperation (WPEC) Subgroup 26 [10], have allowed one to quantify the required accuracies to meet most of the design target accuracies of Table I, and Table IV gives a summary of such requirements. This assessment has been made possible by the availability of covariance data, generated as a result of collaboration led by BNL with LANL and ORNL participation (the so-called BOLNA covariance data set), and by the availability of state-of-the-art sensitivity analysis tools. The availability of reliable covariance data has made it possible to show the significance of the impact of the correlation data (the off-diagonal values of a covariance matrix) as shown in Table V. The uncertainty for the
2728
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE IV: Fast reactor uncertainty reduction requirements to meet design target accuracies. Energy Range CurrentAccuracy (%) TargetAccuracy (%) U-238 σinel 6.07 - 0.498 MeV 10 - 20 2-3 σcapt 24.8 - 2.04 keV 3-9 1.5 - 2 Pu-241 σfiss 1.35 MeV - 454 eV 8 - 20 2-8 Pu-239 σcapt 498 - 2.04 keV 7 - 15 4-7 Pu-240 σfiss 1.35 - 0.498 MeV 6 1.5 - 2 ν 1.35 - 0.498 MeV 4 1-3 Pu-242 σfiss 2.23 - 0.498 MeV 19 - 21 3-5 Pu-238 σfiss 1.35 - 0.183 MeV 17 3-5 Am-242m σfiss 1.35 MeV - 67.4 keV 17 3-4 Am-241 σfiss 6.07 - 2.23 MeV 12 3 Cm-244 σfiss 1.35 - 0.498 MeV 50 5 Cm-245 σfiss 183 - 67.4 keV 47 7 Fe-56 σinel 2.23 - 0.498 MeV 16 - 25 3-6 Na-23 σinel 1.35 - 0.498 MeV 28 4 - 10 Pb-206 σinel 2.23 - 1.35 MeV 14 3 Pb-207 σinel 1.35 - 0.498 MeV 11 3 Si-28 σinel 6.07 - 1.35 MeV 14 - 50 3-6 σcapt 19.6 - 6.07 MeV 53 6
be taken with some precaution. In fact, they have been derived with the method originally proposed by L.N. Usachev [11], which does not account for the effect of data correlations. In this respect, the method can be generalized as follows. In fact, the unknown uncertainty data requirements can be obtained by solving a minimization problem where the sensitivity coefficients, in conjunction with the existing constraints, provide the needed quantities to find the solutions. The following function should be minimized: λi = min bi2 i
TABLE V: keff uncertainties (in % Δkeff /keff ) for Sodiumcooled Fast Reactors with diagonal only terms and full covariance matrix (selected isotopes and total value). Diagonal Full U-238 0.16 0.24 Pu-238 0.34 0.64 Pu-239 0.13 0.19 Pu-240 0.38 0.66 Pu-241 0.52 0.96 Pu-242 0.26 0.41 Am-242m 0.37 0.73 Cm-244 0.27 0.40 Cm-245 0.19 0.39 Fe-56 0.37 0.55 Na-23 0.23 0.25 Total 1.04 1.82
with constraints + 2 + i SRi bi SRi + i,j SRi bi bj cij SRj < QR .
keff of a small size SFR almost doubles when correlations are taken into account. The Subgroup 26 studies have pointed out that “the present uncertainties on the nuclear data should be significantly reduced, in order to get full benefit from the advanced modeling and simulation initiative” [10]. Only a “parallel effort in advanced simulation and in nuclear data improvement” will provide designers with more general and well validated calculation tools that would allow them to meet design target accuracies. A further output of WPEC Subgroup 26 has been the proposal for new entries in the OECD-NEA High Priority Request List, based on uncertainty reduction requirements to meet design target accuracies, as previously indicated. We note, though, that due caution should be exercised in using these outputs, as some of the covariance data have been clearly labeled as being of preliminary nature. As for the target accuracies of Table IV, they have to
The cij are the correlation coefficients of the original data covariance matrix, and the b,i are the unknown variance values needed to meet the requirements. To derive the expression above, we have made the assumption that the correlation terms in the covariance matrix do not change. In that expression the SRi are the sensitivity coefficients of integral parameter R to nuclear data, QR is the target accuracy on the integral parameter R, and λi are cost parameters. This expression can be easily generalized to a set of integral parameters and/or to a set of reference systems. To establish priorities for reducing uncertainties, the b,i values should be compared with the variance data in the original covariance matrices. This helps to decide if an action should be undertaken to meet the requirement. Finally, and apart from the approximation of the expression used to find the required variance reductions, three issues should also be carefully assessed: a) the “representativity” of the reference integral parameters; b) the robustness of the initial covariance data and c) the choice of λi [10].
2729
Needs and Issues... IV.
NUCLEAR DATA SHEETS erence system, it reduces to
COMBINED USE OF COVARIANCE DATA AND OF INTEGRAL EXPERIMENTS A.
M. Salvatores et al.
˜bi = S T B ˜ Ri p SRi 1 ∼ = bi 1 − 2 1 − j,m (rj,m − rj,m t rm,t rj,t ) 2 2 ri,j (1 − rt,m ) ×
Theoretical background
Some of the most important requirements, as shown in Table IV, are difficult to meet using only differential experiments, even if innovative experimental techniques are used. The use of integral experiments has been essential in the past to insure enhanced predictions for power fast reactor cores. A combined use of scientifically based covariance data and of selected integral experiments can be made using classical statistical adjustment techniques (see [12, 13] among many other references). These techniques provide adjusted nuclear data for a wide range of applications, together with new, improved covariance data and bias factors (with reduced uncertainties) for the required design parameters, in order to meet target accuracies. In fact, if we define Bp the a priori nuclear data covariance matrix, SR the sensitivity matrix of the integral design parameters Ri (i=1, . . . , I) to the nuclear data pk (k =1, . . . , K), the a priori covariance matrix of the integral design parameters is given by T BR = SR Bp SR .
It can be shown that, using a set of J integral experiments A, characterized by a sensitivity matrix SA , besides a set of statistically adjusted cross section data, a new (a ˜p can be obtained [11, 12]. posteriori) covariance matrix B As a result of the statistical adjustment procedure, one can then evaluate the resulting reduction of uncertainty matrix for the different design parameters Ri (i =1, . . . , I) as follows: ˜R = S T B ˜ B R p SR = T T T Bp SR )−1 (SA Bp SA + BA )−1 (SA Bp SR )2 , BR 1 − (SR ˜p is the new covariance matrix for the nuclear where B data and SR is the sensitivity matrix of the set of I design parameters in the reference system, to the K nuclear data. SA is the sensitivity matrix of the set of J integral experiments. This very general equation has to be used to fully understand the impact of a data adjustment on the performances of a well defined reference system. Moreover, one can use the same equation to understand the effectiveness of a data adjustment and its “extrapolability” to a set of different reference systems. For this purpose, one has to introduce in the previous equation as matrix SR the sensitivity matrix of the design parameters Rin (i=1, . . . , I; n=1, . . . , N) of a set of N reference systems, to the K nuclear data. When the previous equation is applied to the case of only one integral parameter i (e.g., the keff ) of the ref-
j
− 2
t,m
ri,m ri,t (rt,m − rj,t rj,m )
,
m,t
where the r are correlation coefficients
ri,j
Si Bp SjT = 1/2 1/2 , Si Bp SiT Sj Bp SjT
rj,m
Sm Bp SjT = . T )1/2 S B S T 1/2 (Sm Bp Sm j p j
In the case of only two experiments (i.e., J=2) one obtains
2 1 2 ˜bi ∼ ri,1 + ri,2 − 2 ri,1 ri,2 r1,2 . = bi 1 − 2 1 − r1,2 From these expressions it is clear that in case of only one reference design parameter, if at least one experiment is such that Si ∼Sj , one reaches with that experiment alone the maximum uncertainty reduction. If both experiments have “unsatisfactory” sensitivity profiles, one can still have an uncertainty reduction if r1,2 ∼0. These are fairly intuitive conditions, but, as indicated previously, only the full use of the formulation can result in reduced uncertainties for the full set of design parameters of the reference system. The so-called “representativity” of one single experiment with respect to one single design parameter, is of little use to meet the requirement of reducing the uncertainties of a set of design parameters for one (or more) reference system(s).
B.
A GNEP-oriented application
A new initiative in this field is a GNEP/DOE sponsored 3-years Project with the participation of ANL, BNL, INL, and LANL. The scope of the project is to produce a set of improved nuclear data using improved covariance data and a carefully selected set of integral experiments. As far as covariance data is concerned, they will be produced from thermal energy to 20 MeV in a 33-energy group representation for 65 priority materials including actinides, structural materials and fission products. The covariance data are provided for elastic, inelastic, capture
2730
Needs and Issues...
NUCLEAR DATA SHEETS
TABLE VI: List of integral experiments to be used in the statistical adjustment. Experiment Parameter to be an- Fuel Type Pu/ alyzed (U+Pu) Crit. React. Irrad. mass Rates Exp. GODIVA Yes Yes U Metal 0.0 JEZEBEL239 Yes Yes Pu Metal 1.0 JEZEBEL240 Yes Pu Metal 1.0 ZPR-3/53 Yes Yes PuC-UC 0.42 ZPR-3/54 Yes Yes PuC-UC 0.42 ZPPR-15 Yes Yes Pu-U Met 0.13 COSMOa Yes PuO2-UO2 0.27 CIRANOa Yes Yes PuO2-UO2 0.27 PROFILb Yes PuO2-UO2 0.27 TRAPUb Yes PuO2-UO2 0.27
M. Salvatores et al.
TABLE VIII: Sensitivity of keff to mubar (mean scattering cosine) of selected isotopes. Isotope System Fission Capture Elastic Inelastic Mubar O-16 ABRa) -0.002 -0.027 -0.004 EFR -0.003 -0.018 -0.008 Na-23 ABRa) -0.002 -0.001 -0.009 -0.005 EFR -0.001 0.015 -0.005 -0.009 Fe-56 ABR a) -0.023 -0.020 -0.036 -0.013 EFR -0.011 0.036 -0.018 -0.016 LFR -0.010- 0.019 -0.017 -0.008 U-238 ABR a) 0.065 -0.198 0.022 -0.042 -0.010 EFR 0.074 -0.144 0.044 -0.032 -0.022 LFR 0.060 -0.127 0.029 -0.038 -0.015 Pb LFR -0.009 0.124 -0.029 -0.021 a) Oxide fuel
a) Experiment in the MASURCA facility [15] b) Irradiation experiment in the PHENIX reactor [16]
• S(α,β) thermal scattering data (mostly for thermal or thermalized neutron systems), TABLE VII: keff Uncertainties (pcm) calculated with BOLNA and Adjusted Covariance data for Advanced Burner Reactors. Reactor BOLNA Adjusted 4 groups Covariance ABR Oxide 1438 639 ABR Metal 1460 639
and (n,2n) cross sections, while for the actinides, fission cross sections and nubar values are also provided. The entire nuclear data covariance activities are coordinated by BNL, with support from LANL in the range of actinides and light nuclei. The resulting covariance data are utilized by ANL and INL for analysis. A preliminary result, based on the selection of a limited number of existing, well documented (QA-grade) integral experiments (see Table VI and Ref. [13]), is shown in Table VII [14]. That table shows how the resulting covariance matrix (collapsed to 4 energy groups) after adjustment allows one to reduce the initial uncertainty on the keff of two different reference design configurations, i.e., an ABR with metal fuel and an ABR with oxide fuel.
V.
OTHER DATA UNCERTAINTY NEEDS
The previous discussion has been focused on neutron cross section data needs. However, there are needs for uncertainty data in a wider range of nuclear data. Some typical cases are indicated below: • Fission spectrum uncertainty data, as pointed out by Subgroup 26, and several papers at this Workshop, • Photon production data uncertainty (of importance, e.g., for local heating in the core),
• Mubar (mean scattering cosine) uncertainty. For this last case, we show in Table VIII the sensitivity of the keff of several systems to mubar data of selected isotopes. The sensitivity to mubar is compared to the sensitivity of the other nuclear data (fission, capture, elastic and inelastic). These results show that the orders of magnitude are often not negligible, and comparable to those of other important data. As an example, if we fold these sensitivity coefficients with an uncertainty of, e.g., 10%, we obtain uncertainties on the keff values of approximately 300 pcm in a system like the European Fast Reactor (EFR) and even larger for a Lead-cooled Fast Reactor (LFR). These effects are essentially due to the role of mubar in the leakage component, as it can easily be seen remembering that leakage the L is proportional (e.g., in the diffusion approximation) to B 2 /{3[Σa + Σs (1 − μ)]}.
VI.
CONCLUSIONS
Uncertainty, sensitivity, nuclear data adjustments and validation studies are essential components of any significant effort in simulation improvement. These studies can be credible and can be used in a convincing and effective manner to perform, e.g., design optimization and to assess safety features and design margins, only if the data uncertainty information is of the highest quality, reliable and science based. The worldwide ongoing efforts are an answer to a longstanding need of reactor physics. The expected results will greatly help to promote a new generation of innovative systems with ambitious goals as far as sustainability, waste minimization, proliferation risk reduction, together with further improved safety and economy, as required, e.g., by the Generation-IV objectives.
2731
Needs and Issues...
NUCLEAR DATA SHEETS
[1] G. Aliberti et al., “Nuclear Data Sensitivity, Uncertainty and Target Accuracy Assessment for Future Nuclear Systems,” Annals of Nuclear Energy, vol. 33, 700-733, 2006. [2] M. Salvatores, G. Aliberti, G. Palmiotti, “Nuclear Data Needs for Advanced Reactor Systems. A NEA Nuclear Science Committee Initiative,” ND 2007: Int. Conf. on Nuclear Data for Science and Technology, Nice, France, April 22-27, 2007. [3] D. Rochman, M. Herman, P. Obloˇzinsk´ y and. S. F. Mughabghab, “Preliminary Cross-Section Covariances for WPEC Subgroup 26,” Tech. Rep. BNL-77407-2007IR, Brookhaven National Laboratory, 2007. [4] G. Palmiotti et al., “Requirements for Advanced Simulation of Nuclear Reactor and Chemical Separation Plants,” ANL-AFCI-168, Argonne National Laboratory, May 2006. [5] H. Khalil, R. Hill, “Evaluation of Liquid-Metal Reactor Design Options for Reduction of the Sodium Void Worth”, Nucl. Sci. Eng. Vol. 109, 221, 1991. [6] M. Salvatores, R. Hill, I. Slessarev, G. Youinou, “The Physics of TRU Transmutation – A Systematic Approach to the Intercomparison of Systems,” Proc. PHYSOR 2004 – The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments, Chicago, Illinois, April 25-29, on CD-ROM, American Nuclear Society, Lagrange Park, Il., 2004. [7] G. Aliberti, G. Palmiotti, M. Salvatores, “Spectral Effects at Core-Reflector Interface in Fast Neutron Systems” Trans. ANS Annual Meeting, vol. 88, 516, June 2003, San Diego, California. [8] G. Palmiotti, M. Salvatores, “Use of Integral Experiments in the Assessment of Large Liquid-Metal Fast Breeder Reactor Basic Design Parameters,” Nucl. Sci.
M. Salvatores et al.
Eng., vol. 87, 333, 1984. [9] C. Chabert, et al.,“An improved method for fuel cycle analysis at equilibrium and its application to the study of fast burner reactors with variable conversion ratio”, Proc. Int. Conf. PHYSOR 2008, Interlaken, Switzerland (to be published). [10] M. Salvatores et al., “OECD/NEA WPEC Subgroup 26 Final Report: Uncertainty and Target Accuracy Assessment for Innovative Systems Using Recent Covariance Data Evaluations”, Report NEA/WPEC-26, Paris 2008. [11] L. N. Usachev and Y. Bobkov, “Planning on Optimum Set of Microscopic Experiments and Evaluations to Obtain a Given Accuracy in Reactor Parameter Calculations”, Report INDC CCP-19U, IAEA, Vienna 1972. [12] A. Gandini and M. Petilli, “AMARA: A Code Using the Lagrange Multipliers Methods of Nuclear Data Adjustment,” Report RT/FI (73)39, CNEN, Italy, 1973. [13] M. Salvatores, G. Aliberti and G. Palmiotti, “Nuclear Data Validation and Fast Reactor Design Performances Uncertainty Reduction,” Trans. ANS Meeting, 96, 519, Boston, Massachusetts, June 22-26, 2007. [14] G. Palmiotti et al., “A Global Approach to the Physics Validation of Simulation Codes for Future Nuclear Systems”, Proc. Int. Conf. PHYSOR 2008 (to be published). [15] “Benchmark on Computer Simulation of MASURCA Critical and Subcritical Experiments”, Report NEA/NSC/DOC (2005)23, Paris 2005. [16] A. D’Angelo, F. Cleri, P. Marimbeau, M. Salvatores and. J. P. Grouiller, “Analysis of Sample and Fuel Pin Irradiation in PHENIX for Basic Nuclear Data Validation”, Nucl. Sci. and Eng., vol. 105, 244, 1990.
2732
Needs and Issues of Covariance Data Application M. Salvatores,1,2,3∗ G. Palmiotti,1 G. Aliberti,2 H.Hiruta,1 R. McKnight,2 P. Obloˇzinsk´ y,4 W.S. Yang2 1
Idaho National Laboratory, NSE Division, 2525 Fremont Ave. P.O. Box 1625, Idaho Falls, ID 83415, USA 2 Argonne National Laboratory, NE Division, Argonne, IL 60439, USA 3 CEA-Cadarache, 13108 St-Paul-Lez-Durance, France and 4 National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA (Received July 28, 2008) Needs and issues related to covariance data application are provided and several examples are given to show their impact on important design parameters that affects both the economic and the safety of advanced nuclear reactors.
I.
DESIGN TARGET UNCERTAINTIES
Recently, extensive sensitivity and uncertainty studies [1, 2] and the availability of new neutron cross section covariance data [3] have allowed the preliminary quantification of the impact of current nuclear data uncertainties on the design parameters of the major Gen-IV systems. This, in particular, holds for sodium-cooled fast reactors (SFRs) with different fuels (oxide or metal), fuel composition (e.g., different Pu/TRU ratios) and different conversion ratios† . These studies have pointed out that present uncertainties on the nuclear data should be significantly reduced, in order to obtain the full benefit from advances in modeling and simulation. Only a parallel effort in advanced simulation and in nuclear data improvement will be able to provide designers with more general and well validated calculation tools to meet design target accuracies. Current as well as targeted uncertainties for some of the most important SFR design parameters have been assessed [4], applicable to the Advanced Burner Reactor (ABR) design. A typical example, related to core neutronics, is given in Table I. This table gives, for each parameter, the respective contribution to the current estimated uncertainties of both input data and simulation tools. It is also important to realize that covariance data can have a significant impact on innovative design features, as it will be illustrated in the next Section. Then, in subsequent two Sections we will first discuss uncertainties and target accuracies, followed by explanation of the combined used of covariance data and integral experiments.
∗ Electronic
address: [email protected] [†] Gen-IV stands for new generation of nuclear power reactors. TRU means transuranium materials.
0090-3752/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.nds.2008.11.001
II.
PHYSICS ISSUES AND IMPACT OF UNCERTAINTIES
A wide range of systems has been investigated, both within the Advanced Fuel Cycle Initiative (AFCI) and the Global Nuclear Energy Partnership (GNEP) initiatives, even if no final design choices have been made yet. Some expected new significant features (for the core and fuel cycle) depend heavily on nuclear data knowledge and uncertainties. Moreover, design margins, optimization and choice of options will depend on an accepted accuracy level in simulations. This is an important issue, since economy and safety are at stake. Typical examples of nuclear data dependent innovative design features are: • Cores with low reactivity loss during the cycle, • Cores with increased inventory of Minor Actinides (MA) in the fuel, • Cores with no uranium blankets. Both core design and the associated fuel cycle features have to be considered.
A.
Optimization of Na void reactivity versus Internal Breeding Gain
In order to optimize the irradiation time, any neutronically efficient design will have a low reactivity loss during the cycle Δρ, i.e., an Internal Breeding Gain, IBG, close to zero Δρ ≈ {(νσf − σa )U 238 − (νσf − σa )P u239 } × IBG cycle −(νσf − σa )U 238 + σcF P ,
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE I: Current and targeted uncertainties for some Sodium-cooled Fast Reactors (SFR) design parameters. Neutronics: Core Parameter Current uncertainty Targeted Parameter Current uncertainty Targeted Uncertainty Uncertainty Input Data Origin Modeling Input Data Origin Modeling (A Priori) Origin (A Priori) Origin Multiplication Fac1.5% 0.5% 0.3% Reactivity Co7% 15% 7% tor, (Δkeff /keff ) efficients: Total Power 1% 3% 2% Reactivity 20% 20% 10% Peak Coefficients: Component Power 1% 6% 3% Fast Flux for 7% 3% 3% Distribution Damage Conversion Ratio 5% 2% 2% Kinetics 10% 5% 5% (Absolute Value) Parameters Control Rod Worth: 5% 6% 5% Local Nuclide 5% 3% 2% Element Densities: Major Actinides Control Rod Worth: 5% 4% 2% Local Nu30% 10% 10% Total clide Densities: Minor Actinides Burnup Reactivity 0.7% 0.5% 0.3% Fuel Decay Heat 10% 3% 5% Swing (Δkeff /keff ) at Shutdown 6 5 4
Eta
where ν is the number of neutrons per fission, σf is the fission cross section and σa is the absorption cross section. A reactivity loss close to zero allows for the designing of very long life cores, with evident economic benefits. Moreover, such design requires an excess reactivity at the beginning of cycle near zero, avoiding potentially large reactivity insertions (control rod run out, handling error, etc.), which can represent a potential safety issue. However, it has been demonstrated in the past [5] that, in the case of Na-cooled Fast reactors, there is a very tight correlation between the reactivity loss per cycle and the reactivity associated with the coolant voiding of the core. It is also well known that in the case of the Na-cooled Fast Reactors, it is important to assess the sign and the magnitude of the reactivity coefficient associated to a coolant voiding of the core. This reactivity coefficient can be expressed, e.g. using diffusion perturbation theory, as follows
Cm245 3 2 1 0 1.E+0
U238
Pu239
Am241 Pu238 1.E+1
Cm244 1.E+2
1.E+3
1.E+4
Np237 1.E+5
1.E+6
1.E+7
1.E+8
Energy [eV]
FIG. 1: η = ν σf /σa energy shape for selected actinides.
related to the change in self shielding of e.g. 238 U, due to the Na density variation and in general ASelfSh is positive and small. As indicated above, the sign of the spectral component 1 Δkeff c {NNa = σNa,j Φj Φ+ SNa is determined by the energy shape of the adjoint flux, j keff F j which, at first approximation, is related to the energy + + dependence of η = νσf /σa . − NNa σNa (j → k)Φj (Φk − Φj ) Since different fissile isotopes (239 Pu, 235 U, 233 U) and j,k Minor Actinides have significantly different η shapes (see + a Fig. 1), the result is very different adjoint flux shapes, } δDj ∇Φj ∇Φ+ dV − N δσ Φ Φ − i i,j j j j according to the actual TRU composition of the core, in j i j particular at high energy, and, as a consequence, different = ANa − SNa − L − ASelfSh. values of the (positive) Na void scattering component. As for the spectral component SNa one can have SNa >0 The trade-off between the Na void coefficient and the + or <0 according to the sign of (Φ+ Φ ). The leakage Δρ/cycle plays an important role in core design (e.g., to j k component L is such that L>0 and its absolute value will define number and position of control rods in the core). increase with the decrease of the geometrical dimensions. In the case of a core with a low Δρ/cycle, the spectral Finally, the absorption component ANa is positive and, component becomes dominant, according to the qualitain general, relatively small. The ASelfSh component is tive scheme indicated in Fig. 2. 2726
Needs and Issues... ĭ+
NUCLEAR DATA SHEETS
Adjoint flux behavior behavior as Adjoint functionflux of E: as fonction of E:
) ~
20
M. Salvatores et al.
Variation %
Variation (%)
A
Increase of positive Na void reactivity coefficient
10 A
Q6 f 6 a DB 2
0 B
- 10
D C D
- 20
High U-238/Pu-239 ratio
Low U-238/Pu-239 ratio
A : Vidange sodium B : Doppler C : Burn-up D : Bêta effectf
C
- 30 B
- 40 Act. Mineurs
Chargés % MA in (%) fuel
- 50
Positive Na void reactivity coeff.
Decrease of reactivity loss/cycle
0
Negative Na void reactivity coeff.
2
Decrease of ȕeff
FIG. 2: Schematic adjoint flux energy shape and Na void reactivity/IBG relation.
The presence of a high content of Minor Actinides in future innovative fast reactors is related to the specific mission assigned to fast reactors in order to minimize the amount of waste produced by a power fleet. The content of MA in the fuel varies according to specific strategies. For example, fast neutron cores that should stabilize the MA inventory in the fuel cycle within a TRU homogeneous recycle strategy, will have a MA content of 2-5% and definitely less than 10% (see e.g. Ref. [6]). A “burner” fast reactor can have MA content in the fuel which varies according to the conversion ratio and the type of fuel. For example, fuel with no uranium could accommodate up to 50% MA. However, the physics characteristics of most MA isotopes (e.g., low or very low fraction of delayed neutrons, threshold fission, etc.), will have an impact on key reactivity coefficients, as shown in figure 3, related to an SFR. The features shown in that figure indicate that there can be limitations in the amount of MA that can be added safely to the fuel, because of the need to keep an acceptable compromise between the deterioration of some reactivity coefficients (e.g., Na-void and Doppler coefficients) and the improvement e.g., of the reactivity loss/cycle. Once more, all these feature result from a very delicate balance of effects of different sign, and data uncertainties can play a very significant role, both in a design opti-
6
8
10
Max MA content ~ 2.5% for large size SFR (EFR) and 5-10 % for medium/small size (Phenix)
Decrease of Doppler reactivity coefficient
FIG. 3: Variation of SFR parameters with minor actinide (MA) content in the fuel.
As a consequence, there can be a significant potential impact of nuclear data uncertainties on core feasibility and on its safety assessment. For example, the good compromise on IBG value and Na-void reactivity based on nominal values of both parameters could be revised to take into account uncertainties with practical consequences on, e.g., the core burn-up: an IBG more negative should be found in order to have a less positive Na-void coefficient, with a higher reactivity loss/cycle and consequently a potential reduction of the burn up.
B. Fast reactor cores with high content of Minor Actinides: impact on reactivity coefficients (case of a SFR)
4
TABLE II: keff sensitivity (%) of 56 Fe data. Experiment/ Type Capture Elastic Inelastic Configuration ZPR3-53 Blanket -0.11 0.76 -0.02 ZPR3-54 Reflector -1.40 16.5 1.50 CIRANO Reflector -1.50 6.28 -0.24 ZPPR-15 Blanket -1.54 1.67 -2.55 ABR-Metal Reflector -1.49 3.05 -3.06 ABR Oxide Reflector -1.73 1.79 -3.29
Total 0.63 16.6 4.55 2.43 -1.51 -3.23
mization phase and in more advanced design phases, e.g. during the safety case assessment. For example, if one wants to limit a) the increase of the Na-void reactivity, b) the decrease of the Doppler effect and c) the decrease of the delayed effective neutron fraction, one has to limit the amount of MA in the fuel, e.g. to 3-5%; however, nuclear data uncertainties have to be accounted for, and they can suggest a more conservative choice of the MA content. This impact can be more significant for “burner” cores with potentially a higher MA content in the fuel.
C.
Blanket versus reflector in fast reactors
Nonproliferation concerns suggest avoiding uranium blankets in the next generation FRs, and replacing them with reflectors. However, the presence of reflectors induces neutron spectrum transients at the core/reflector interface that need specific calculation methods to be described appropriately [7]. It has been shown that the spectrum transient effects are also sensitive to data issues [7]. For example, the keff sensitivity to the different Fe-56 data, in configurations with a blanket or with a reflector is indicated in Table II. Very different sensitivities are shown, as expected. Also in this case, nuclear data uncertainties will have to be taken into account, both in keff and in the power distribution close to the interface or on the peak-to-average
2727
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE III: Isotopic contributions to the decay heat (MeV/s/g) at fuel unloading (case of metal fuel and Pu/MA=0.5 in the feed) for two values of Conversion Ratio. CR=0.0 CR=1.0 Total decay heat 3.08 E+12 3.53 E+10 Pu-238 component 3.00 E+11 6.50 E+9 Am-241 component 5.23 E+10 1.62 E+9 Cm-242 component 6.64 E+11 1.35 E+10 Cm-244 component 2.03 E+12 1.03E+10 %Cm-244 in the fuel 8.63 E-02 5.2E-04
R= 186 cm R= 115 cm R= 73 cm R= 59 cm
S 1 /(1 O1 / O2 ) Probable range for ABR cores
FIG. 4: Spatial sensitivity profile of a central control rod in different fast reactors.
power ratio in the core, with potential consequences on the core optimization (e.g., worth and position of external control rods).
D.
topes, with different decay constants, give comparable contributions. Their build-up during irradiation should be known rather accurately. This means there is a need for accurate capture and fission data in the related Bateman equations, in order to avoid uncertainties greater than about 10% on isotope densities, as indicated in Table I.
III. UNCERTAINTIES AND TARGET ACCURACIES: LESSONS LEARNED WITH WPEC SUBGROUP 26
Control rod spatial sensitivity to nuclear data uncertainties
It is well known that space dependent effects (e.g. the system response to a localized perturbation) are dependent on core size or, more precisely, on the Boltzmann equation eigenvalue separation [8]. Due to this very fundamental feature, the control rod worth in a fast reactor (whatever its coolant) is affected differently by cross section uncertainties according to the size of the core, control rod location and environment. In Fig. 4, the spatial dependence of the sensitivity of a control rod worth is shown in cores of different sizes, each characterized by the separation of the first two eigenvalues of the related Boltzmann operator. In the probable range of future ABR cores, the sensitivity of control rod worth (both for isolated rods but even more for a control rod system in interaction) to data uncertainty can be rather significant and should be accounted for in the core safety assessment.
E.
Some fuel cycle issues
As for the Fuel Cycle, in an assessment of the most important features for the Fuel Cycle, it is important to have a clear understanding (i.e., with low uncertainty) of isotope contributions to specific effects. This drives the related design requirement of Table I. For example, it can be shown [9] that the decay heat at fuel unloading differs by a factor of about 100 for fast reactor cores with a Conversion Ratio CR=0 or =1. The relative isotopic contributions at fuel unloading are shown in Table III for a typical ABR related fuel cycle. In the case with CR=0, Cm-244 is practically the only contributor, while, in the case with CR=1, several iso-
Recent work to assess uncertainties on a wide range of integral parameters and on a wide range of systems, has been performed within an international initiative and a final report is being issued [10]. The studies, that have been performed on a wide range of e.g. fast reactors with different coolants, fuel type or Pu/MA ratios, allow an analysis in more details of some of the effects previously discussed. In particular, uncertainties have been derived for keff , Na-void reactivity, Δρ/cycle, nuclide densities at the end of irradiation etc. For most of the physics phenomena indicated above, the level of uncertainty is high enough to jeopardize optimization and to require introducing costly safety margins. If preliminary design calculations are certainly possible with current nuclear data, the type of uncertainties that have been pointed out will certainly have an important impact on more advanced design phases. In particular, several target accuracies as indicated in Table I will not be met. The studies, performed in the OECD/NEA Working Party on Evaluation Cooperation (WPEC) Subgroup 26 [10], have allowed one to quantify the required accuracies to meet most of the design target accuracies of Table I, and Table IV gives a summary of such requirements. This assessment has been made possible by the availability of covariance data, generated as a result of collaboration led by BNL with LANL and ORNL participation (the so-called BOLNA covariance data set), and by the availability of state-of-the-art sensitivity analysis tools. The availability of reliable covariance data has made it possible to show the significance of the impact of the correlation data (the off-diagonal values of a covariance matrix) as shown in Table V. The uncertainty for the
2728
Needs and Issues...
NUCLEAR DATA SHEETS
M. Salvatores et al.
TABLE IV: Fast reactor uncertainty reduction requirements to meet design target accuracies. Energy Range CurrentAccuracy (%) TargetAccuracy (%) U-238 σinel 6.07 - 0.498 MeV 10 - 20 2-3 σcapt 24.8 - 2.04 keV 3-9 1.5 - 2 Pu-241 σfiss 1.35 MeV - 454 eV 8 - 20 2-8 Pu-239 σcapt 498 - 2.04 keV 7 - 15 4-7 Pu-240 σfiss 1.35 - 0.498 MeV 6 1.5 - 2 ν 1.35 - 0.498 MeV 4 1-3 Pu-242 σfiss 2.23 - 0.498 MeV 19 - 21 3-5 Pu-238 σfiss 1.35 - 0.183 MeV 17 3-5 Am-242m σfiss 1.35 MeV - 67.4 keV 17 3-4 Am-241 σfiss 6.07 - 2.23 MeV 12 3 Cm-244 σfiss 1.35 - 0.498 MeV 50 5 Cm-245 σfiss 183 - 67.4 keV 47 7 Fe-56 σinel 2.23 - 0.498 MeV 16 - 25 3-6 Na-23 σinel 1.35 - 0.498 MeV 28 4 - 10 Pb-206 σinel 2.23 - 1.35 MeV 14 3 Pb-207 σinel 1.35 - 0.498 MeV 11 3 Si-28 σinel 6.07 - 1.35 MeV 14 - 50 3-6 σcapt 19.6 - 6.07 MeV 53 6
be taken with some precaution. In fact, they have been derived with the method originally proposed by L.N. Usachev [11], which does not account for the effect of data correlations. In this respect, the method can be generalized as follows. In fact, the unknown uncertainty data requirements can be obtained by solving a minimization problem where the sensitivity coefficients, in conjunction with the existing constraints, provide the needed quantities to find the solutions. The following function should be minimized: λi = min bi2 i
TABLE V: keff uncertainties (in % Δkeff /keff ) for Sodiumcooled Fast Reactors with diagonal only terms and full covariance matrix (selected isotopes and total value). Diagonal Full U-238 0.16 0.24 Pu-238 0.34 0.64 Pu-239 0.13 0.19 Pu-240 0.38 0.66 Pu-241 0.52 0.96 Pu-242 0.26 0.41 Am-242m 0.37 0.73 Cm-244 0.27 0.40 Cm-245 0.19 0.39 Fe-56 0.37 0.55 Na-23 0.23 0.25 Total 1.04 1.82
with constraints + 2 + i SRi bi SRi + i,j SRi bi bj cij SRj < QR .
keff of a small size SFR almost doubles when correlations are taken into account. The Subgroup 26 studies have pointed out that “the present uncertainties on the nuclear data should be significantly reduced, in order to get full benefit from the advanced modeling and simulation initiative” [10]. Only a “parallel effort in advanced simulation and in nuclear data improvement” will provide designers with more general and well validated calculation tools that would allow them to meet design target accuracies. A further output of WPEC Subgroup 26 has been the proposal for new entries in the OECD-NEA High Priority Request List, based on uncertainty reduction requirements to meet design target accuracies, as previously indicated. We note, though, that due caution should be exercised in using these outputs, as some of the covariance data have been clearly labeled as being of preliminary nature. As for the target accuracies of Table IV, they have to
The cij are the correlation coefficients of the original data covariance matrix, and the b,i are the unknown variance values needed to meet the requirements. To derive the expression above, we have made the assumption that the correlation terms in the covariance matrix do not change. In that expression the SRi are the sensitivity coefficients of integral parameter R to nuclear data, QR is the target accuracy on the integral parameter R, and λi are cost parameters. This expression can be easily generalized to a set of integral parameters and/or to a set of reference systems. To establish priorities for reducing uncertainties, the b,i values should be compared with the variance data in the original covariance matrices. This helps to decide if an action should be undertaken to meet the requirement. Finally, and apart from the approximation of the expression used to find the required variance reductions, three issues should also be carefully assessed: a) the “representativity” of the reference integral parameters; b) the robustness of the initial covariance data and c) the choice of λi [10].
2729
Needs and Issues... IV.
NUCLEAR DATA SHEETS erence system, it reduces to
COMBINED USE OF COVARIANCE DATA AND OF INTEGRAL EXPERIMENTS A.
M. Salvatores et al.
˜bi = S T B ˜ Ri p SRi 1 ∼ = bi 1 − 2 1 − j,m (rj,m − rj,m t rm,t rj,t ) 2 2 ri,j (1 − rt,m ) ×
Theoretical background
Some of the most important requirements, as shown in Table IV, are difficult to meet using only differential experiments, even if innovative experimental techniques are used. The use of integral experiments has been essential in the past to insure enhanced predictions for power fast reactor cores. A combined use of scientifically based covariance data and of selected integral experiments can be made using classical statistical adjustment techniques (see [12, 13] among many other references). These techniques provide adjusted nuclear data for a wide range of applications, together with new, improved covariance data and bias factors (with reduced uncertainties) for the required design parameters, in order to meet target accuracies. In fact, if we define Bp the a priori nuclear data covariance matrix, SR the sensitivity matrix of the integral design parameters Ri (i=1, . . . , I) to the nuclear data pk (k =1, . . . , K), the a priori covariance matrix of the integral design parameters is given by T BR = SR Bp SR .
It can be shown that, using a set of J integral experiments A, characterized by a sensitivity matrix SA , besides a set of statistically adjusted cross section data, a new (a ˜p can be obtained [11, 12]. posteriori) covariance matrix B As a result of the statistical adjustment procedure, one can then evaluate the resulting reduction of uncertainty matrix for the different design parameters Ri (i =1, . . . , I) as follows: ˜R = S T B ˜ B R p SR = T T T Bp SR )−1 (SA Bp SA + BA )−1 (SA Bp SR )2 , BR 1 − (SR ˜p is the new covariance matrix for the nuclear where B data and SR is the sensitivity matrix of the set of I design parameters in the reference system, to the K nuclear data. SA is the sensitivity matrix of the set of J integral experiments. This very general equation has to be used to fully understand the impact of a data adjustment on the performances of a well defined reference system. Moreover, one can use the same equation to understand the effectiveness of a data adjustment and its “extrapolability” to a set of different reference systems. For this purpose, one has to introduce in the previous equation as matrix SR the sensitivity matrix of the design parameters Rin (i=1, . . . , I; n=1, . . . , N) of a set of N reference systems, to the K nuclear data. When the previous equation is applied to the case of only one integral parameter i (e.g., the keff ) of the ref-
j
− 2
t,m
ri,m ri,t (rt,m − rj,t rj,m )
,
m,t
where the r are correlation coefficients
ri,j
Si Bp SjT = 1/2 1/2 , Si Bp SiT Sj Bp SjT
rj,m
Sm Bp SjT = . T )1/2 S B S T 1/2 (Sm Bp Sm j p j
In the case of only two experiments (i.e., J=2) one obtains
2 1 2 ˜bi ∼ ri,1 + ri,2 − 2 ri,1 ri,2 r1,2 . = bi 1 − 2 1 − r1,2 From these expressions it is clear that in case of only one reference design parameter, if at least one experiment is such that Si ∼Sj , one reaches with that experiment alone the maximum uncertainty reduction. If both experiments have “unsatisfactory” sensitivity profiles, one can still have an uncertainty reduction if r1,2 ∼0. These are fairly intuitive conditions, but, as indicated previously, only the full use of the formulation can result in reduced uncertainties for the full set of design parameters of the reference system. The so-called “representativity” of one single experiment with respect to one single design parameter, is of little use to meet the requirement of reducing the uncertainties of a set of design parameters for one (or more) reference system(s).
B.
A GNEP-oriented application
A new initiative in this field is a GNEP/DOE sponsored 3-years Project with the participation of ANL, BNL, INL, and LANL. The scope of the project is to produce a set of improved nuclear data using improved covariance data and a carefully selected set of integral experiments. As far as covariance data is concerned, they will be produced from thermal energy to 20 MeV in a 33-energy group representation for 65 priority materials including actinides, structural materials and fission products. The covariance data are provided for elastic, inelastic, capture
2730
Needs and Issues...
NUCLEAR DATA SHEETS
TABLE VI: List of integral experiments to be used in the statistical adjustment. Experiment Parameter to be an- Fuel Type Pu/ alyzed (U+Pu) Crit. React. Irrad. mass Rates Exp. GODIVA Yes Yes U Metal 0.0 JEZEBEL239 Yes Yes Pu Metal 1.0 JEZEBEL240 Yes Pu Metal 1.0 ZPR-3/53 Yes Yes PuC-UC 0.42 ZPR-3/54 Yes Yes PuC-UC 0.42 ZPPR-15 Yes Yes Pu-U Met 0.13 COSMOa Yes PuO2-UO2 0.27 CIRANOa Yes Yes PuO2-UO2 0.27 PROFILb Yes PuO2-UO2 0.27 TRAPUb Yes PuO2-UO2 0.27
M. Salvatores et al.
TABLE VIII: Sensitivity of keff to mubar (mean scattering cosine) of selected isotopes. Isotope System Fission Capture Elastic Inelastic Mubar O-16 ABRa) -0.002 -0.027 -0.004 EFR -0.003 -0.018 -0.008 Na-23 ABRa) -0.002 -0.001 -0.009 -0.005 EFR -0.001 0.015 -0.005 -0.009 Fe-56 ABR a) -0.023 -0.020 -0.036 -0.013 EFR -0.011 0.036 -0.018 -0.016 LFR -0.010- 0.019 -0.017 -0.008 U-238 ABR a) 0.065 -0.198 0.022 -0.042 -0.010 EFR 0.074 -0.144 0.044 -0.032 -0.022 LFR 0.060 -0.127 0.029 -0.038 -0.015 Pb LFR -0.009 0.124 -0.029 -0.021 a) Oxide fuel
a) Experiment in the MASURCA facility [15] b) Irradiation experiment in the PHENIX reactor [16]
• S(α,β) thermal scattering data (mostly for thermal or thermalized neutron systems), TABLE VII: keff Uncertainties (pcm) calculated with BOLNA and Adjusted Covariance data for Advanced Burner Reactors. Reactor BOLNA Adjusted 4 groups Covariance ABR Oxide 1438 639 ABR Metal 1460 639
and (n,2n) cross sections, while for the actinides, fission cross sections and nubar values are also provided. The entire nuclear data covariance activities are coordinated by BNL, with support from LANL in the range of actinides and light nuclei. The resulting covariance data are utilized by ANL and INL for analysis. A preliminary result, based on the selection of a limited number of existing, well documented (QA-grade) integral experiments (see Table VI and Ref. [13]), is shown in Table VII [14]. That table shows how the resulting covariance matrix (collapsed to 4 energy groups) after adjustment allows one to reduce the initial uncertainty on the keff of two different reference design configurations, i.e., an ABR with metal fuel and an ABR with oxide fuel.
V.
OTHER DATA UNCERTAINTY NEEDS
The previous discussion has been focused on neutron cross section data needs. However, there are needs for uncertainty data in a wider range of nuclear data. Some typical cases are indicated below: • Fission spectrum uncertainty data, as pointed out by Subgroup 26, and several papers at this Workshop, • Photon production data uncertainty (of importance, e.g., for local heating in the core),
• Mubar (mean scattering cosine) uncertainty. For this last case, we show in Table VIII the sensitivity of the keff of several systems to mubar data of selected isotopes. The sensitivity to mubar is compared to the sensitivity of the other nuclear data (fission, capture, elastic and inelastic). These results show that the orders of magnitude are often not negligible, and comparable to those of other important data. As an example, if we fold these sensitivity coefficients with an uncertainty of, e.g., 10%, we obtain uncertainties on the keff values of approximately 300 pcm in a system like the European Fast Reactor (EFR) and even larger for a Lead-cooled Fast Reactor (LFR). These effects are essentially due to the role of mubar in the leakage component, as it can easily be seen remembering that leakage the L is proportional (e.g., in the diffusion approximation) to B 2 /{3[Σa + Σs (1 − μ)]}.
VI.
CONCLUSIONS
Uncertainty, sensitivity, nuclear data adjustments and validation studies are essential components of any significant effort in simulation improvement. These studies can be credible and can be used in a convincing and effective manner to perform, e.g., design optimization and to assess safety features and design margins, only if the data uncertainty information is of the highest quality, reliable and science based. The worldwide ongoing efforts are an answer to a longstanding need of reactor physics. The expected results will greatly help to promote a new generation of innovative systems with ambitious goals as far as sustainability, waste minimization, proliferation risk reduction, together with further improved safety and economy, as required, e.g., by the Generation-IV objectives.
2731
Needs and Issues...
NUCLEAR DATA SHEETS
[1] G. Aliberti et al., “Nuclear Data Sensitivity, Uncertainty and Target Accuracy Assessment for Future Nuclear Systems,” Annals of Nuclear Energy, vol. 33, 700-733, 2006. [2] M. Salvatores, G. Aliberti, G. Palmiotti, “Nuclear Data Needs for Advanced Reactor Systems. A NEA Nuclear Science Committee Initiative,” ND 2007: Int. Conf. on Nuclear Data for Science and Technology, Nice, France, April 22-27, 2007. [3] D. Rochman, M. Herman, P. Obloˇzinsk´ y and. S. F. Mughabghab, “Preliminary Cross-Section Covariances for WPEC Subgroup 26,” Tech. Rep. BNL-77407-2007IR, Brookhaven National Laboratory, 2007. [4] G. Palmiotti et al., “Requirements for Advanced Simulation of Nuclear Reactor and Chemical Separation Plants,” ANL-AFCI-168, Argonne National Laboratory, May 2006. [5] H. Khalil, R. Hill, “Evaluation of Liquid-Metal Reactor Design Options for Reduction of the Sodium Void Worth”, Nucl. Sci. Eng. Vol. 109, 221, 1991. [6] M. Salvatores, R. Hill, I. Slessarev, G. Youinou, “The Physics of TRU Transmutation – A Systematic Approach to the Intercomparison of Systems,” Proc. PHYSOR 2004 – The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments, Chicago, Illinois, April 25-29, on CD-ROM, American Nuclear Society, Lagrange Park, Il., 2004. [7] G. Aliberti, G. Palmiotti, M. Salvatores, “Spectral Effects at Core-Reflector Interface in Fast Neutron Systems” Trans. ANS Annual Meeting, vol. 88, 516, June 2003, San Diego, California. [8] G. Palmiotti, M. Salvatores, “Use of Integral Experiments in the Assessment of Large Liquid-Metal Fast Breeder Reactor Basic Design Parameters,” Nucl. Sci.
M. Salvatores et al.
Eng., vol. 87, 333, 1984. [9] C. Chabert, et al.,“An improved method for fuel cycle analysis at equilibrium and its application to the study of fast burner reactors with variable conversion ratio”, Proc. Int. Conf. PHYSOR 2008, Interlaken, Switzerland (to be published). [10] M. Salvatores et al., “OECD/NEA WPEC Subgroup 26 Final Report: Uncertainty and Target Accuracy Assessment for Innovative Systems Using Recent Covariance Data Evaluations”, Report NEA/WPEC-26, Paris 2008. [11] L. N. Usachev and Y. Bobkov, “Planning on Optimum Set of Microscopic Experiments and Evaluations to Obtain a Given Accuracy in Reactor Parameter Calculations”, Report INDC CCP-19U, IAEA, Vienna 1972. [12] A. Gandini and M. Petilli, “AMARA: A Code Using the Lagrange Multipliers Methods of Nuclear Data Adjustment,” Report RT/FI (73)39, CNEN, Italy, 1973. [13] M. Salvatores, G. Aliberti and G. Palmiotti, “Nuclear Data Validation and Fast Reactor Design Performances Uncertainty Reduction,” Trans. ANS Meeting, 96, 519, Boston, Massachusetts, June 22-26, 2007. [14] G. Palmiotti et al., “A Global Approach to the Physics Validation of Simulation Codes for Future Nuclear Systems”, Proc. Int. Conf. PHYSOR 2008 (to be published). [15] “Benchmark on Computer Simulation of MASURCA Critical and Subcritical Experiments”, Report NEA/NSC/DOC (2005)23, Paris 2005. [16] A. D’Angelo, F. Cleri, P. Marimbeau, M. Salvatores and. J. P. Grouiller, “Analysis of Sample and Fuel Pin Irradiation in PHENIX for Basic Nuclear Data Validation”, Nucl. Sci. and Eng., vol. 105, 244, 1990.
2732