Study of Predictor-corrector methods. for Monte Carlo Burnup Codes. Dan Kotlyar Dr. Eugene Shwageraus. Supervisor

Transcription

1 Serpent International Users Group Meeting Madrid, Spain, September 19-21, 2012 Study of Predictor-corrector methods for Monte Carlo Burnup Codes By Supervisor Dan Kotlyar Dr. Eugene Shwageraus

2 Introduction High Conversion LWRs Eigen value D (no T-H) Time, days 1

3 Introduction High Conversion LWRs Eigen value D (T-H) 2D (no T-H) Time, days 1

4 Introduction Advisor s response Power share in seed What? T [i] T[i+1] What? Why? Programming? Physical (model)? Numerical (coupling)? Height, cm Cause How? Solution! Is this problem common for all MC based codes? 2

5 Outline Simplify the model Generic PWR assembly in 3D No T-H feedback, but non-uniform coolant density fixed in time Study efficient schemes for coupling MC with multi-physics feedbacks Identify origins of the observed problem Focus of this work: Examine MC-burnup coupling schemes (& t size) Was this problem identified earlier? Recent burnup schemes sensitivity studies Description of the test case (3D) & Results Conclusions & future work 3

6 Introduction Examples of Integrated MC-burnup codes: MOCUP (Moore et al., 1995) MCODE (Xu et al., 2002) MONTEBURNS (Trellue, 2003) SERPENT (Leppänen, 2007) BGCore (Fridman et al., 2008) MCNPX (Hendricks et al., 2008) What about Integration of MC-burnup-TH BGCore SERPENT Issues related to Integrated MC codes Accuracy Computation requirements 4

7 Objectives Investigate stability and accuracy of current depletion calculations Monte-Carlo/Burnup coupling scheme Depletion timestep size Commonly used methods Explicit Euler predictor method Euler predictor-corrector method Propose an extension to the modified Euler predictor-corrector Accuracy as the predictor-corrector method Computation requirements as the predictor method 5

8 Possible multi-physics coupling schemes Integration scheme 1 Beginning of Step analysis (predictor) Inner loop: TH-neutronic Total exe time (Depletion/TH) n T mcnp Outer loop: Depletion Integration scheme 2 Inner loop: Depletion Outer loop: TH-neutronuc Important issues Combination of #1 and #2 Calculation time Source convergence Distribution of errors Variable convergence tolerances Neutronics MC (r, E) Depletion module New concentration for next step Temp & density Distribution Thermal feedback Power & BU Distribution (x n) (x 1) 6

9 Recent sensitivity studies Yamamoto et al., 2008 Projected predictor corrector method Linear correlation between the number density and the microscopic RR Tested for the Gd-bearing 2D fuel assembly Carpenter et al., 2010 (Bettis Atomic Power Laboratory) Modified Log Linear correlation (Yamamoto) Tested for the Gd-bearing 2D fuel assembly Isotalo et al., 2011 Higher order methods Use more (previous) BU points Tested on PWR pin cell and seed/blanket assembly Saadi et al., 2012 Burnup sensitivity analysis Tested on 1D UO2 PWR unit cell Dufek et al., 2009 Numerical stability of MC-burnup codes Instability demonstrated on infinitely reflected 3D unit cell 7

10 Burnup coupling scheme methodology Formulation of the burnup problem Explicit Euler Predictor Euler Predictor-Corrector Extended Predictor-Corrector

11 Burnup calculations n n+1 n+2 EOC Eigen-value transport problem Burnup equation Matrix Exponential solution L N r, t FN r, t r,, E, t0 dn r, t dt N 0 1 k r, E, tede Nr, t r N r t t n n n1 1 exp n 8

12 Burnup calculations n n+1 n+2 EOC Eigen-value equation Burnup equation L N r, t FN r, t r,, E, t0 dn r, t dt 0 1 k r, E, tede Nr, t Mat. Exp. solution N r N r t t n n n1 1 exp n Objective: coupled space-energy-time dependent solution: Coupling scheme: Independent neutronics and depletion solvers 8

13 Euler predictor-corrector methods 1. Explicit Euler Predictor method Reaction rate (RR) calculation (neutronics - ) at BOT Depletion with BOT RR s N n+1 N n const RR n N n+1 n n+1 const RR n+1 Time/BU 2. Modified Euler Predictor-Corrector method Reaction rate calculation ( ) with predicted N n+1 values Depletion with EOT (n+1) reaction rates to obtain corrected N n+1 values Finally, the predicted and corrected ND are averaged Used as initial values for the following step 9

14 Why is the P-C method is not enough? t! Power Share Normalized commulative energy (50 days) Quote: Predictor-corrector methods are numerically explicit The stable size may not be much larger than of that of the predictor Numerical methods for engineers and scientists, Joe D. Hoffman 0 50 d t E i t 0,50 P t t Predictor Corrector 50 days d 10 d 25 d 50 d Height, cm Height, cm 10

15 Suggested E-P-C method 3. Extended Predictor-Corrector method (E-P-C) Assume (power, RR, σ, N) are known for the previous [t n-1,t n ] interval Deplete with previous average RR N n+0.5 ( =N n exp[-rr n-1 t 0.5 ] ) Update transport solution (RR calculation ) at MOT (t n+0.5 ) Deplete with MOT RR s N n+1 Reaction rate calculation ( ) with predicted N n+1 values Depletion with EOT (n+1) reaction rates to obtain corrected N n+1 values Finally, the predicted and corrected ND are averaged N n-1 N n N n+0.5 const RR n+0.5 N n+1 n-1 75d n 100d n d const RR n+1 n+1 150d Time/BU 11

16 Different coupling scheme results

17 3D assembly test case Radial +20,000 layout Cylindrical pins With guide tubes Axial layout / Coolant density profile 21.5 cm Coolant Density, gr/cm 3 12

18 BGCore vs. SERPENT comparison Concentration of Pu239, #/b cm Eigen value Parameter k-eff Maximum difference, SERPENT vs. BGCore ~70 pcm Xe 135 ~0.5 % ~0.4 % BGCore SERPENT U 235 ~0.4 % Pu E E E-05 BGCore SERPENT Time, days 7.0E E E E E-05 Method : predictor-corrector Timestep : 5 days 2.0E E E Time, days 13

19 Burnup coupling scheme method & T Case no Method Predictor Predictor-Corrector a Predictor-Corrector Predictor-Corrector Description Extended-Predictor-Corrector a. Chosen to be the reference case T, days Designated P-C-0 (25d) P-C-1 ( 5d) P-C-1 (25d) P-C-1 (50d) E-P-C (50d) 14

20 Difference in eigen value, pcm Burnup coupling scheme method & T Case no Method Predictor Predictor-Corrector a Predictor-Corrector Predictor-Corrector Description Extended-Predictor-Corrector a. Chosen to be the reference case T, days Designated P-C-0 (25d) P-C-1 ( 5d) P-C-1 (25d) P-C-1 (50d) E-P-C (50d) P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) Time, days 14

21 Concentration of Xe-135 Difference in eigen value, pcm Burnup coupling scheme method & T Case no. Method Description T, days Designated P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) 1 Predictor 25 P-C-0 (25d) Predictor-Corrector a 5 P-C-1 ( 5d) Predictor-Corrector 25 P-C-1 (25d) Predictor-Corrector 50 P-C-1 (50d) 0 5 Extended-Predictor-Corrector 50 E-P-C (50d) Time, days a. Chosen to be the reference case 1.2E E E E E E E+00 P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) P-C-1 (5d) Time, days 14

22 Flux distribution, n/s cm2 Concentration of Xe-135 Difference in eigen value, pcm Burnup coupling scheme method & T Case no. Method Description T, days Designated P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) 1 Predictor 25 P-C-0 (25d) Predictor-Corrector a 5 P-C-1 ( 5d) Predictor-Corrector 25 P-C-1 (25d) Predictor-Corrector 50 P-C-1 (50d) 0 5 Extended-Predictor-Corrector 50 E-P-C (50d) Time, days a. Chosen to be the reference case 4.0E E+14 T(i) T(i+1) 1.2E E E E E E E E E Height, cm 8.0E E E E E+00 P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) P-C-1 (5d) Time, days 14

23 Eigen value The effect of non-uniform coolant density Eigen value Realistic coolant density along the z-axis Averaged coolant density along the z-axis P-C-0 (25d) P-C-1 (25d) P-C-1 (50d) P-C-0 P-C Time, days Time, days 15

24 k-eff Comparison between P-C and E-P-C Difference [pcm] to P-C-1-25d Concentration of U-235 Relative difference to P-C-1-25d Maximum difference (compared to P-C-1) in different core parameters Parameter k-eff Maximum difference, P-C-1 (50d) vs. P-C-1 (25 d) ~500 pcm Maximum difference, E-P-C (50d) vs. P-C-1 (25 d) ~20 pcm Xe 135 ~2.0 % ~0.5 % U 235 ~1.9 % ~0.1 % Pu 239 ~2.5 % ~0.2 % P-C-1 (25d) P-C-1 (50d) E-P-C (50d) DIFF (P-C-1) DIFF (E-P-C) E E E E % 0.5% 0.0% -0.5% E % Time, days 6.0E E E E-04 P-C-1 (25d) P-C-1 (50d) E-P-C (50d) DIFF (P-C-1) DIFF (E-P-C) Time, days -1.5% -2.0% -2.5% -3.0% 16

25 Conclusions & future work

26 Summary Sensitivity studies of different coupling schemes were performed Explicit Euler predictor method may be unstable for some 3D problems Predictor-corrector method with Δt ~50d does not resolve the problem EPC method allows using larger timesteps Maintains sufficient accuracy Future work: How/When is the Thermal-hydraulic feedback should be applied? 17

27 Thank You for your attention!