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ORNL Seminar 19.05.11 Reactor Point Kinetics-- Then and Now Barry D. Ganapol Fellow

ORNL Seminar 19.05.11 Reactor Point Kinetics-- Then and Now Barry D. Ganapol Fellow Advanced Institute of Studies Unibo and DIENCA Visiting Professor. and University of Arizona UTK. Some thoughts concerning PKE algorithms--. v Errors and missing information in papers

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ORNL Seminar 19.05.11 Reactor Point Kinetics-- Then and Now Barry D. Ganapol Fellow

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  1. ORNL Seminar 19.05.11 Reactor Point Kinetics--Then and Now Barry D. Ganapol Fellow Advanced Institute of Studies Unibo and DIENCA Visiting Professor and University of Arizona UTK

  2. Some thoughts concerning PKE algorithms-- v Errors and missing information in papers v Unsubstantiated claims of accuracy, simplicity, usefulness and elegance v Lack of benchmarks and benchmarking strategy v The “simple” algorithm is missing v Extreme accuracy has always been achievable v No ultimate PKE algorithm currently exists

  3. Sketch of Point Kinetics Equations

  4. Separability: N(t) Adjoint weighting: dN(t) N(t)+ m +S(t) N(t)- i = 1,…,m

  5. A Survey of Past Solutions to the PKEs

  6. Nuclear Reactor Kinetics by G.R. Keepin, 1965 + Integral form (RTS Code 1960) + Laplace transform and inversion + Requires extensive tables of poles and residuals + RTS code advanced at the time + Too difficult to use routinely

  7. A New Solution of the Point Kinetics Equations J. A. W. daNóbrega NSE 46, 366-375 (1971) v Consider constant reactivity insertion and constant source

  8. Unnecessarily complicated for outcome v Require inverse of A which is argued to be too computationally expensive (at the time) - Advocates Padé approximant, e.g., P(2,0) Note: All eigenvalues are not necessary but still require extreme eigenvalues v At best 5x10-5 relative error

  9. Solution of the Reactor Kinetics Equations by Analytical Continuation John Vigil NSE 29, 292-401 (1967) Taylor Series Recurrence

  10. + Continuous Analytical Continuation Time Discretization + Time step control + A method ahead of its time

  11. On the Numerical Solution of the Point Kinetics Equations by Generalized Runge-Kutta Methods J. Sanchez NSE 103, 94-99 (1989) All coefficients are specified

  12. Method Underperforms L = 2e-05s

  13. A new integral method for solving the point reactor neutron kinetics equations Li, Chen, Luo, Zhu, ANE 36, 427-432 (2009) v Start from integral equation and assume

  14. Method performs poorly L = 2e-05s

  15. Aboanber Methods: PWS: an efficient code system for solving space-independent nuclear reactor dynamics A.E. Aboanber*, Y.M. Hamada Annals of Nuclear Energy 29 (2002) 2159–2172

  16. Claimed “exact” solution is not so Ramp

  17. Solution of the point kinetics equations in the presence of Newtonian temperature feedback by Pad´e approximations via the analytical inversion method A E Aboanber and A ANahla J. Phys. A: Math. Gen. 35 (2002) 9609–9627 Same method as daNóbrega and Sanchez

  18. Inaccurate at large time

  19. A Resolution of the Stiffness Problem of Reactor Kinetics Y. Chao, A. Attard NSE 90, 40-46 (1985) Define: Choose u and w to confine most variation to N

  20. Confusing mathematical physics modeling

  21. An analytical solution of the point kinetics equations with time-variable reactivity by the decomposition method Claudio Z. Petersen , Sandra Dulla, Marco T.M.B. Vilhena, PieroRavetto Progress in Nuclear Energy xxx (2011) Adomian Decomposition Method

  22. Inappropriate claims of accuracy, utility and simplicity L = 10-6s

  23. ….hence…. A simple reliable, robust algorithm to solve the PKEs is lacking. + Must think numerically and use - new computational architectures - robust numerical methods - experimental numerical methods + Must abandon the outmoded ideas of time step control and the “minimum time step competition”.

  24. A Survey of New Solutions to the PKEs + GPCA + TS

  25. mGPCA G(?)Piecewise Constant Approximation to the Solution of Point Kinetics Eqns (PKEs) Recall:

  26. v First consider constant reactivity insertion (without source)

  27. DiagonalizeA : A = UWU-1 + Eigenvalues from + Eigenvectors form U from + U-1 = VT from transpose + W = diag{wk;k=1,…,G} Attributable to daNobrega, Sanchez, Allen, Aboanber

  28. In-hour Equation:

  29. + Exact solution for step insertion - Algorithm Summary HQR Algorithm for wk Explicit eigenvector representation Note: All done through linear algebra

  30. v Now consider prescribed reactivity insertion Efficient numerical solution of the point kinetics equations in nuclear reactor dynamics M. Kinard and E. Allen ANE 31 1039-1051 (2004) - Note: must introduce a time step and solve for eigenvalues for each time interval

  31. An Implicit Method for Solving the Lumped Parameter Reactor-Kinetics Equations by Repeated Extrapolation M. Izumi and T Noda NSE 41 299-303 (1970) Combined R-K with repeated Richardson extrapolation to improve the FD/RK scheme Another article method ahead of its time Can this concept be generalized ?

  32. v Convergence acceleration + True solution based on the limit + Form a sequence of solutions

  33. + Apply a convergence accelerator to to find a new sequence such that is found from the asymptotic behavior (in n) of original sequence Some accelerators are: Romberg, Aitkin Wynn-epsilon (W-e), Euler transformation

  34. +GPCA Ganapolized Piecewise Constant Reactivity Approximation - Goal is extreme accuracy tj-1 tj Sequentially halve interval Build a sequence of solutions over all grids Accelerate convergence via Romberg or W-e Begin each interval with converged IC

  35. Ramp 0.1b/s For all edits

  36. $0.50 Step Insertion in a fast reactor L = 10-7,10-8,….,10-19 L =10-19 L = 10-7

  37. Claudio Z. Petersen , Sandra Dulla, Marco T.M.B. Vilhena, PieroRavetto Progress in Nuclear Energy xxx (2011) Ramp GPCA

  38. 1965 2011

  39. Here is robust for ramp considered earlier GPCA Correct to 9-places in comparison with FD (Ganapol/NSE Letter)

  40. v Test by manufactured solution + Solve for reactivity + Specify N(t) + Input imply

  41. Assume exponential to power level Implies

  42. Error Measure for all 3 cases e = 10-10

  43. mTaylor series (TS) solution to PKEs (J. Vigil/1967) + GPCA - requires discretization solution and In-hour - not analytical - iteration to include non-linear reactivity + TS solution most natural solution and gives an analytical solution

  44. + Form TS in interval [tj-1,tj] + Naturally generate the following recurrence:

  45. + Numerical implementation - Must proceed with caution TS slowly converging and therefore sensitive to round off (from “swell”) - Use Continuous Analytical Continuation (CAC) Choose interval [tj-1,tj] to limit number of terms in TS to K Accelerate convergence of partial sums via W-e at original and added time edits if necessary

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