Perturbation/sensitivity calculations with Serpent

Transcription

1 SERPENT workshop Cambridge, September 2014 Perturbation/sensitivity calculations with Serpent Manuele Auero, Adrien Bidaud, Pablo Rubiolo LPSC/CNRS Grenoble

2 Calculating the complete β e...coupling CFD and neutron transport in the MSFR uncertainty in MSRs Nuclear data + DNP decay constants + uid ow

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4 Considered response functions Eect of a perturbation of the parameter x on the response R : S R x dr/r dx/x Considered response functions: R = k e Eective multiplication factor R = R = Σ 1, φ Σ 2, φ φ, Σ 1 φ (Simple extension of IFP method, briey presented last φ Bilinear ratios (Adjoint-weighted quantities), Σ 2 φ R =? Something else

5 Particle's weight perturbation All the cross sections (and probability distributions) are articially increased by a factor f. Events are rejected with a probability of (1 1/f ). f = 0.5 w w 0 ( 1 + dσ n,2n Σ n,2n ) ( 1 + dσs Σ s ) ( 1 dσ f Σ f ( 1 + dσ f Σ f ) ) ( ( 1 + dσs Σ s 1 + dσs Σ s ) ( 1 dσc Σ c ) ) ( 1 + dσ f Σ f )...

6 The general idea... Adopting the distribution of the corrected particles weight in the reference system as unbiased estimator of the exact neutron ux distribution in the perturbed system. Re-normalization of the total population weight. Convergence of the propagation (latent?) generations.

7 Particle's weight perturbation x = nuclear data for reaction r, on the isotope i, in the material m, in the incident neutron energy bin e, in the volume s (outgoing neutron energy bin e and scattering cosine bin l ) w n x/x α ( ) (n,g) w n ACCx (n,g) REJ x g=(α λ) α = present generation λ = number of propagation generations ACC x = accepted events x in the history of the particle n REJ x = rejected events x

8 (method) R = Σ 1, φ Σ 2, φ R = Σ 1 + Σ 1, φ + φ Σ 2 + Σ 2, φ + φ Neglecting cross terms... R R S R x = = Σ 1, φ Σ 1, φ Σ 2, φ Σ 2, φ + Σ 1, φ Σ 1, φ Σ 2, φ Σ 2, φ Σ2 Σ1, φ, φ x/x x/x + Σ 1, φ Σ 2, φ }{{} direct terms Σ 1, φ x/x Σ 2, φ x/x Σ 1, φ Σ 2, φ }{{} indirect terms

9 (method) Considering track-length estimators (for simplicity)... Σ 1, φ = q Σ 1, φ x/x n α w n l t Σ 1 t n = q n α t n w n w n/w n x/x l t Σ 1 Σ 1, φ x/x = q [ α w n n α t n g=(α λ) ( ACC (n,g) x ) ] REJ (n,g) x l tσ 1

10 (method) Indirect terms: Σ 1, φ x/x Σ 1, φ = [ α ( q w n ACC (n,g) x n α t n g=(α λ) q n α t n w n l t Σ 1 ) ] REJ (n,g) x l t Σ 1 Average net number of x events (i.e., real - virtual) in the last λ generations, weighted on the contributions to the track length estimator of Σ 1, φ Indirect part of S R x is obtained as the dierence between the average number of net x events in the last λ generations, weighted on the tally contributions for two generic detectors Σ 1, φ and Σ 2, φ

11 (results) R = φ(r, E) σ 238 U f (E) de d r φ(r, E) σ 235 U f (E) de d r Jezebel (Pu sphere) PU-MET-FAST U/ 235 U ssion rate ratio (measured in the center of the system)

12 (results) Flattop-Pu (Popsy) PU-MET-FAST U/235 U ssion rate ratio (measured in the center of the system) (not in scale) Manuele Au ero - LPSC/CNRS Grenoble

13 (results) R = φ(r, E) σ 238 U f (E) de d r φ(r, E) σ 235 U f (E) de d r UAM TMI-1 PWR pin-cell 238 U/ 235 U ssion rate ratio in the fuel pellet

14 (results) ERANOS results from Sandro Jezebel - F28/F25 - Pu elastic scattering F28/F25 sensitivity - 10 generations 0,02 Sensitivity per lethargy unit 0,01 0-0,01-0,02 Extended SERPENT-2 (JEFF-3.1) Extended SERPENT-2 (ENDF/B-VII) ERANOS TSUNAMI-1D -0,03-0, Energy (ev)

15 (results) ERANOS results from Sandro Jezebel - F28/F25 - Pu inelastic scattering F28/F25 sensitivity - 10 generations 0-0,02 Sensitivity per lethargy unit -0,04-0,06-0,08-0,1 Extended SERPENT-2 (JEFF-3.1) Extended SERPENT-2 (ENDF/B-VII) ERANOS TSUNAMI-1D -0, Energy (ev)

16 (results) ERANOS results from Sandro U-238 Fission cross-section Cross-section (b) JEFF-3.1 ENDF/B-VII Energy (MeV)

17 (results) ERANOS results from Sandro Popsy (Flattop) - F28/F25 - Pu chi total F28/F25 sensitivity - 10 generations - ENDF/B-VII 0,3 Sensitivity per lethargy unit 0,2 0,1 0-0,1 Extended SERPENT-2 TSUNAMI-1D -0,2-0, Energy (ev)

18 (results) ERANOS results from Sandro Popsy (Flattop) - F28/F25 - Pu fission F28/F25 sensitivity - 10 generations - JEFF Sensitivity per lethargy unit Extended SERPENT-2 ERANOS Energy (ev)

19 (results) ERANOS results from Sandro 0,1 UAM TMI-1 PWR cell - F28/F25 - H - total F28/F25 sensitivity - 10 generations - ENDF/B-VII 0 Sensitivity per lethargy unit -0,1-0,2-0,3-0,4 Extended SERPENT-2 TSUNAMI-1D -0,5-0, Energy (ev)

20 (results) ERANOS results from Sandro UAM TMI-1 PWR cell - F28/F25 - U disappearance F28/F25 sensitivity - 10 generations - ENDF/B-VII 0.15 Sensitivity per lethargy unit Extended SERPENT-2 TSUNAMI-1D Difference (S-T) Energy (ev)

21 (results) ERANOS results from Sandro Energy integrated sensitivity coecients for Jezebel for the response function R = F28/F25. x S R x JEFF-3.1 ENDF/B-VII Serpent Eranos Rel. di Serpent TSUNAMI-1D Rel. di 239 Pu σ tot ± 0.1% % ± 0.1% % 239 Pu σ inl ± 0.0% % ± 0.0% % 239 Pu σ ela ± 0.2% % ± 0.2% % 239 Pu σ s ± 0.1% % ± 0.1% % 240 Pu σ tot ± 0.3% % ± 0.3% % 239 Pu σ dis ± 0.1% % ± 0.1% % 240 Pu σ inl ± 0.2% % ± 0.2% % 240 Pu σ ela ± 0.7% % ± 0.7% % 239 Pu σn,xn ± 0.2% % ± 0.2% % 240 Pu σ s ± 1.9% % ± 0.7% % 240 Pu σ dis ± 0.4% % ± 0.5% % 241 Pu σ tot ± 1.6% %?? ± 1.5% % 241 Pu σ inl ± 0.9% % ± 0.8% %

22 Bilinear ratios (method) R = φ, Σ 1 φ φ, Σ 2 φ Examples: β e = α coolant = φ 1, χ d ν d Σ f φ k e l e = φ 1, χ t ν t Σ f φ k e φ, Σ t,coolantφ φ, 1 χ t ν t Σ f φ k e φ, φ, 1 φ v 1 χ t ν t Σ f φ k e

23 Bilinear ratios (method) R = φ + φ, (Σ 1 + Σ 1 ) (φ + φ) φ + φ, (Σ 2 + Σ 2 ) (φ + φ) R φ, Σ 1 φ φ, Σ 2 φ φ, Σ 1 φ φ, Σ 2 φ φ, Σ 1 φ φ, Σ 2 φ = R φ, Σ 1 φ φ +, Σ 2 φ φ, Σ 1 φ φ +, Σ 2 φ φ, Σ 1 φ φ, Σ 2 φ φ Σ 1, φ φ Σ 2, φ φ φ, Σ 1 φ φ, Σ 2 S R x/x x/x x/x x/x x = φ, Σ 1 φ φ +, Σ 2 φ φ, Σ 1 φ φ +, Σ 2 φ + φ, Σ 1φ x/x φ, Σ 1 φ φ, Σ 2φ x/x φ, Σ 2 φ

24 Bilinear ratios (method) Adopting Iterated Fission Probability importance estimators: I (γ) n = 1 q 1 w n k d (γ) n w k Importance of neutrons in generation α is calculated as function of the neutron descendants in generation α + γ Eect of perturbation on neutron importance: I (γ) n x/x = 1 w n k d (γ) n w k x/x 1 w n 2 w n x/x k d (γ) n w k

25 Bilinear ratios (method) Indirect terms... Eect of perturbation on the forward ux: φ φ, Σ 1 = w n 1 l tσ 1 x/x n α t n x/x w n k d (γ) n w k Eect of perturbation on the adjoint ux: φ, Σ 1 φ = w n l tσ 1 1 x/x n α t n w n k d (γ) n w k x/x 1 w n 2 w n x/x k d (γ) n w k Sum of indirect terms (rewritten as function of neutrons in generation α + γ ): φ, Σ 1 φ x/x φ + x/x, Σ 1 φ [ = k (α+γ) w k ( t ( γ) k l tσ 1 ) w k /w k x/x ]

26 Bilinear ratios (method) Example: eective prompt lifetime φ, 1v φ R = φ 1, χ t ν tσ f φ k e Simple IFP estimator for the numerator: φ, 1v φ = 1 q w k ( γ) l k k (α+γ) Numerator terms of the perturbation: φ, 1 φ φ, 1 (α+γ) ( φ w (n,g)accx k (n,g) ) REJ x ( γ) l k v x/x x/x v k (α+γ) g=(α λ) φ, 1v + φ φ, 1v = φ w k ( γ) l k k (α+γ)

27 Bilinear ratios (method) Denominator terms: φ, F φ (α+γ) φ F φ (, φ, Fφ w (n,g)accx k (n,g) ) REJ x x/x x/x x/x k (α+γ) g=(α λ) φ +, Fφ φ +, Fφ φ =, Fφ w k k (α+γ) We nally obtain the sensitivity coecient for l e : history E ( γ) l (ACCx REJ x ) S l history e x = [ ] E (ACCx REJ x ) = E ( γ)l = history COV ( γ) l, (ACCx REJ x ) [ ] E ( γ)l

28 Bilinear ratios (method) S R x = Everything is much more simple... If the quantity R can be estimated as the ratio of two generic Monte Carlo responses R = E [e 1] E [e 2 ] the sensitivity coecient of R with respect to x can be obtained as: COV [ e 1, ] history (ACCx REJ x) E [e 1] COV [ e 2, ] history (ACCx REJ x) E [e 2]

29 Bilinear ratios (results) Jezebel - Leff - Pu elastic scattering Effective prompt lifetime sensitivity generations - ENDF/B-VII 0.08 Extended SERPENT-2 TSUNAMI-1D (EGPT) Sensitivity per lethargy unit Energy (ev)

30 Bilinear ratios (results) Jezebel - Leff - Pu disappearance Effective prompt lifetime sensitivity generations - ENDF/B-VII 0 Sensitivity per lethargy unit Extended SERPENT-2 TSUNAMI-1D (EGPT) Energy (ev)

31 Bilinear ratios (results) Popsy (Flattop) - Leff - Pu fission Effective prompt lifetime sensitivity generations - ENDF/B-VII 0,2 0,1 Sensitivity per lethargy unit 0-0,1-0,2-0,3 Extended SERPENT-2 TSUNAMI-1D (EGPT) -0, Energy (ev)

32 Bilinear ratios (results) 0,14 0,12 0,1 Popsy (Flattop) - Leff - U inelastic scattering Effective prompt lifetime sensitivity generations - ENDF/B-VII Extended SERPENT-2 TSUNAMI-1D (EGPT) Sensitivity per lethargy unit 0,08 0,06 0,04 0,02 0-0,02-0, Energy (ev)

33 Bilinear ratios (results) UAM TMI-1 PWR cell - α coolant - U disappearance coolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII Extended SERPENT-2 TSUNAMI-1D Difference (S-T) Sensitivity per lethargy unit Energy (ev)

34 Bilinear ratios (results) UAM TMI-1 PWR cell - α coolant - U nubar total coolant void reactivity coeff. sensitivity - 4 generations - ENDF/B-VII 1 Sensitivity per lethargy unit Extended SERPENT-2 TSUNAMI-1D Difference (S-T) Energy (ev)

35 Scattering distributions The method can be extended to scattering distribution sensitivities: At each scattering event, two pairs of outgoing energy/scattering angle are sampled One is accepted as real event, the other is rejected as virtual Implicit, continuous constraining of the sensitivity proles

36 Scattering distributions 0.2 Jezebel - Pu elastic scattering Sensitivity to scattering cosine in CoM frame (constrained) - 6 generations - ENDF/B-VII 0.1 Sensitivity per cosine width Effective prompt lifetime -- Extended SERPENT-2 Keff -- Extended SERPENT-2 Keff -- MCNP Kedrowski and Brown, Nucl Sci Eng Vol. 174 (2013) Scattering cosine (CoM frame)

37 Scattering distributions Flattop - l eff - elastic scattering cosine in CoM (constr.) Effective prompt lifetime sensitivity - Extended Serpent generations - ENDF/B-VII 0.2 Sensitivity per cosine width U 239 Pu Scattering cosine (CoM frame)

38 Scattering distributions Jezebel, k e sensitivity to elastic scattering distribution

39 Scattering distributions Jezebel, l e sensitivity to elastic scattering distribution

40 Scattering distributions Jezebel, F 28/F 25 sensitivity to elastic scattering distribution

41 Latent generations convergence ERANOS results from Sandro Keff sensitivity coefficient per lethargy unit 0,05 0,04 0,03 0,02 0,01 0-0,01 Flattop (Pu239 configuration) - U238 Inelastic scattering Extended Serpent2 (LPSC version) - adj-weighted sensitivity - 10 latent generations Ext. Serpent 2-0 gen Ext. Serpent 2-1 gen Ext. Serpent 2-2 gen Ext. Serpent 2-3 gen Ext. Serpent 2-10 gen Eranos (P1 S100) Eranos (P5 S100) -0, Energy (ev)

42 Next steps Complete β e uncertainties in MSFR GPT in Serpent/OpenFOAM for multiphysics application Continuous (E and µ) sensitivity to Legendre moments (from implicitly constrained scattering sensitivities) Exact perturbations Coupled depletion/transport sensitivities......and comparisons with Total Monte Carlo approach Doppler eect

43 THANK YOU FOR THE ATTENTION Vue sur l'agglomération Grenobloise depuis le sommet du Moucherotte (Bertrand93) QUESTIONS? SUGGESTIONS? NEW IDEAS?