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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 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 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
Separability: N(t) Adjoint weighting: dN(t) N(t)+ m +S(t) N(t)- i = 1,…,m
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
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
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
Solution of the Reactor Kinetics Equations by Analytical Continuation John Vigil NSE 29, 292-401 (1967) Taylor Series Recurrence
+ Continuous Analytical Continuation Time Discretization + Time step control + A method ahead of its time
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
Method Underperforms L = 2e-05s
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
Method performs poorly L = 2e-05s
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
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
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
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
Inappropriate claims of accuracy, utility and simplicity L = 10-6s
….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”.
A Survey of New Solutions to the PKEs + GPCA + TS
mGPCA G(?)Piecewise Constant Approximation to the Solution of Point Kinetics Eqns (PKEs) Recall:
v First consider constant reactivity insertion (without source)
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
+ Exact solution for step insertion - Algorithm Summary HQR Algorithm for wk Explicit eigenvector representation Note: All done through linear algebra
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
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 ?
v Convergence acceleration + True solution based on the limit + Form a sequence of solutions
+ 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
+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
Ramp 0.1b/s For all edits
$0.50 Step Insertion in a fast reactor L = 10-7,10-8,….,10-19 L =10-19 L = 10-7
Claudio Z. Petersen , Sandra Dulla, Marco T.M.B. Vilhena, PieroRavetto Progress in Nuclear Energy xxx (2011) Ramp GPCA
1965 2011
Here is robust for ramp considered earlier GPCA Correct to 9-places in comparison with FD (Ganapol/NSE Letter)
v Test by manufactured solution + Solve for reactivity + Specify N(t) + Input imply
Error Measure for all 3 cases e = 10-10
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
+ Form TS in interval [tj-1,tj] + Naturally generate the following recurrence:
+ 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