PROPAGATION OF NUCLEAR DATA UNCERTAINTIES IN FUEL CYCLE USING MONTE-CARLO TECHNIQUE

Transcription

1 PROPAGATION OF NUCLEAR DATA UNCERTAINTIES IN FUEL CYCLE CALCULATIONS USING MONTE-CARLO TECHNIQUE C.J. Díez (1), O. Cabellos (1), J.S. Martínez (1) (1) Universidad Politécnica de Madrid (UPM) International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011) Rio de Janeiro, RJ, Brazil, May 8-12, 2011 M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 1/ 24 C.J. Díez c.diez@upm.es carlosavier@denim.upm.es

2 Abstract The uncertainty propagation in fuel cycle calculations due to Nuclear Data (ND) is a important issue for : Present fuel cycles (e.g. high burnup fuel programme) New fuel cycles designs (e.g. fast breeder reactors and ADS) Different error propagation techniques can be used: Sensitivity analysis Response Surface Method Monte Carlo technique Then, in this paper, p it is assessed the impact of ND uncertainties on the decay heat and radiotoxicity in two applications: Fission Pulse Decay Heat calculation (FPDH) Conceptual design of European Facility for Industrial Transmutation (EFIT) The complete set of uncertainty data for cross sections (EAF2007/UN), decay data and fission yield data (JEFF-3.1.1) are processed and used in ACAB code. M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 2/ 24

3 OUTLINE PART I: PART II: Methodology to propagate ND uncertainties using Monte Carlo technique Application of Monte Carlo technique CONCLUSIONS A. Fission Pulse Decay Heat calculation B. EFIT fuel cycle calculation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 3/ 24

4 PART I Methodology to propagate ND uncertainties Goal: To analyse how ND uncertainties are transmitted to response functions dn dt [ ] [ ] [( ) ] eff eff N + ΦN + (γ Φ N = A N λ N = N ( λ,, γ ) = ) fiss i N i 1) Sensitivity / Uncertainty Analysis (S/U) First order Taylor series (linear approximation) 2) Monte Carlo Uncertainty Analysis (MC) To treat the global effect of all nuclear data uncertainties Without any approximation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 4/ 24

5 PART I Methodology to propagate ND uncertainties Monte Carlo technique Individual / All together sampling (λ,, γ) PDFs Normal distribution ib ti LogNormal distribution ib ti N[ 0, DST ( )] ε N(0, Δ ) I f Δ Maybe < 0 Always i > 0 1 log m ( / ) M 10 ( ) / m0 N(0, M ) Nuclear Data libraries Collapsed Mean Values Uncertainties (Standard Desv) LogNormal distribution λ, γ, Samplig ACAB Results λ γ, λ λ 1, 1 1 2, γ 2, 2 N 1 N n, γ n, n N n 2 M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 5/ 24

6 PART I Methodology to propagate ND uncertainties Uncertainty data Cross section from activation-oriented nuclear data libraries e.g.: W 180 (n,γ) EAF2007-UN E i+1 (ev) i+1( ) W -180N,G E E E E E E E E E E E E E E E E Δ 2 I=1,EAF E i (ev) (relative error, Δ)~ Δ I=1,EXP = Δ I=1,EAF /3 Fission yield from evaluated nuclear data library γ Th232 H 3,400KeV + 1 γ Th KeV H3 JEFF E E E E E E E E E E E E E E M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 6/ 24.

7 PART I Methodology to propagate ND uncertainties Processing and collapsing of nuclear data Collapsing method: -Cross section: Conservation of reaction rate Rate eff i = i ( E) φ( E de = i E ) φ T -Uncertainties: Using Sandwich rule (Propagation of Momentum, first order) Δ 2 = ω T Vω. H. Hiruta et al., Few Group Collapsing of Covariance Matrix Data Based on a Conservation Principle, Nuclear Data Sheets, vol. 109, , (Dec 2008) Collapsing without losing information M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 7/ 24 7

8 PART I Methodology to propagate ND uncertainties Processing and collapsing of nuclear data Cross section Given V the G-by-G variance matrix of the relative cross sections vector, the variance Δ 2 of the relative spectrum-averaged cross section is: with ω = φ1 1 φg G T [, L, ] eff eff φ Fission yield φ Δ 2 φ = φ + φ + L+ eff 1 2 = ω T Vω φ G φ1 1 + φ2 2 + L+ φg G = φ + φ + L+ φ 1 2 G. Given G the M-by-M variance matrix of the relative fission yield vector, the variance Δ 2 of the relative spectrum-averaged fission yield is: with φ 1 ω = [ φ φ 1, fiss G G, fiss, L, eff eff φ fiss fiss ] T where Δ 2 γ = ω T G ω φ γ, i fiss,, i fiss, eff G G γ, i = fiss, fiss, 1 φ G φ G φ G M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 8/ 24

9 PART I Methodology to propagate ND uncertainties Advantages & Disadvantages of Monte Carlo Technique Advantages Collapsing to one energy group Reduce amount of variables to sample No sensitivity coefficients should be calculated No approximation on equations Take into account non-linear effects Disadvantages How to check if the phase space is well sampled? Which PDFs should be taken? Computational demanding M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 9/ 24

10 PART II APPLICATIONS APPLICATIONS: A. Fission Pulse Decay Heat calculation B. EFIT fuel cycle calculation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 10 / 24

11 PART II APPLICATIONS A. Fission Pulse Decay Heat calculation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 11 / 24

12 PART II A. Fission Pulse Decay Heat calculation Description of the problem Decay heat of a single thermal fission event in Pu239 Isotopes: only Fission i Products (FPs) Only Fission yield (FY) and Decay data (Energy/Decay constant) uncertainties are propagated FPs Fission event Pu239 γ Pu239 X 97 Sr DH = λ N E L m Nb41 λβ β E β x 104 Mo42 λβ x β E DH Nb 104 m DH Mo104 β x x L M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 12 / 24

13 PART II A. Fission Pulse Decay Heat calculation Calculations Histories launched/case: 300 Relative error followed Case studied Total decay heat Error in time steps (%) Number of histories Beta decay heat Only known uncertainties // All with uncertainties Gamma decay heat Compared with: - JEFF report 20 - Tobias exp. data For unkown uncertainties Decay Mode Uncertainty Alfa 10% Beta 15% Gamma 15% M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 13 / 24

14 PART II A. Fission Pulse Decay Heat calculation Total decay heat C/E Mean Value JEFF reference value C/E JEFF Tobias 1989 C/E E E E E E E+05 Cooling Time (s) M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 14 / 24

15 PART II APPLICATIONS B. EFIT fuel cycle calculation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 15 / 24

16 PART II B. EFIT fuel cycle calculation Reference system Coolant Pure Lead One of the preliminary conceptual designs of the European Facility for Industrial Transmutation (EFIT) Thermal Power 400 MWth Fuel (Pu, Am)O 2 + MgO Initial mass of actinides tonnes Constant neutron environment: - neutron flux: 3.12 x n/cm 2 s - average energy <E> = 0.37 MeV 1,E-03 1,E-04 1,E-05 Calculations l for discharge burn-up: 1,E GWd/tHM (778 irradiation days) GWd/tHM (3225 irradiation days) 1,E-07 ron Flux Normalized Neut 1,E-08 Initial 400 days Initial total flux intensity = 2.84E+15 n cm -2 s days total flux intensity = 3.12E+15 n cm -2 s -1 1,E-09 1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 Eneutron (MeV) M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 16 / 24

17 PART II B. EFIT fuel cycle calculation Calculations Histories launched: 1000/DH case 300/RTX case Case studied 1. Decay heat 2. Radiotoxicity a.inhalation dose b.ingestion dose step r per each time s (%) relative error Numberof histories All uncertainties are propagated: - Individually, γ, λ - All together ~300 M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 17 / 24

18 PART II B. EFIT fuel cycle calculation Decay heat for 150 GWd/tHM var( y) = DH total DH N i= 1 = i i=isotope var( y ) + i N i, = 1; i Main contributors analysis cov( y, y ) i N var( y i ) >> cov( yi, y ) i = 1 i, = 1; i x 2 x 2 = N i= 1 N 2 2 N y y i i = 2 2 y i x i= 1 error( y ) i 2 y x 2 i 2 var(i)/var( ) cov(i,)/var( ) E E E E E E E E E E+06 Cooling Time (years) M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 18 / 24 18

19 PART II B. EFIT fuel cycle calculation Decay heat for 150 GWd/tHM Main contributors Cm242 Cm244 Pu238 Am241 Pu240 Pu239 TOTAL CM242 CM244 PU238 AM241 PU240 PU239 PO214 PO error (%) E E E E E E E E E E+06 Cooling Time (years) M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 19 / 24

20 PART II B. EFIT fuel cycle calculation Radiotoxicity for 150 GWd/tHM Ingestion Xe133 Cm244 Pu238 Am241 Rn222 Due to FY / XS error Ingestion err ror (%) n error (%) Ingestio ErrorXS+FY+DECAY Error XS Error FY Error DECAY E E E E E E E E E E+06 Cooling time (years) TOTAL XE133 CM244 PU238 AM241 RN E03 1.0E-03 10E02 1.0E-02 10E01 1.0E-01 10E E+00 10E E+01 10E E+02 10E E+03 10E E+04 10E E+05 10E E+06 Cooling time (years) M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 20 / 24

21 CONCLUSIONS M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 21 / 24

22 CONCLUSIONS Monte Carlo technique for ND uncertainty propagation in activation calculations Pre-proccesing of nuclear data is needed: Implemented on ACAB code - Identifying uncertainties - Collapsing of nuclear data Monte Carlo technique VS deterministic calculations / experimental data A good agreement is found between both A method to identify main contributors to error is developed based on MC results PDFs dependency is found in FPDH calculation, but not in EFIT calculation M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 22 / 24

23 ACKNOWLEGMENTS This work is partially supported by: - FP7-EURATOM-FISSION-2009 Proect ANDES/ Ministerio de Educación y Ciencia, SPAIN (Spanish Science and Innovation Ministry) FPU grant AP for the first author. M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 23 / 24

24 THANK YOU FOR YOUR ATTENTION!! M&C 2011 May 8 12,,Rio de Janeiro, RJ, Brazil 24 / 24 C.J. Díez c.diez@upm.es carlosavier@denim.upm.es