Zemskov E.A.
A.I. Leypunsky Institute for Physics and Power Engineering, Obninsk, Russia
The article presents the results of using a sequential matrix approach for analyzing the system of diffusion equations for nuclear reactors in which solutions are constructed with the involvement of a complete system of eigenvectors and eigenvalues.
The criticality calculation reduces to the generalized eigenvalue problem. In addition to the basic solution (Keff and the principal eigenvector), all the dominant relations characterizing the iterative processes are determined. It is shown how such an approach makes it possible to determine the optimal relationship between internal and external iterations.
The matrix representation of the equations of neutron kinetics with allowance for delayed neutrons is treated as a Cauchy problem whose solutions are expressed in terms of a fundamental solution matrix made up of eigenvectors and the exponential operator of the evolution matrix. Knowledge of such solutions allows one to analyze in detail with good accuracy many transient processes, including a pulsed experiment to determine the subcriticality of a medium with fissile materials. These statements are illustrated by calculations of the damping decrement and dynamic reactivity in the model problem.
1. Marchuk G.I. Metody rascheta yadernykh reaktorov [Methods for calculating nuclear reactors]. Moscow, Gosatomizdat Publ., 1961.
2. Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1966.
3. Varga R. Chislennye metody resheniya mnogomernykh mnogogruppovykh diffuzionnykh uravneniy [Numerical methods for solving multidimensional multigroup diffusion equations]. Moscow, Gosatomizdat Publ., 1963.
4. Zemskov E.A., Arzhanov V.I. Spektral'nyy analiz usloviy schetnoy ustoychivosti nekotorykh skhem resheniy zadach prostranstvennoy kinetiki yadernogo reaktora [Spectral analysis of the conditions of countable stability of some schemes for solving the problems of the spatial kinetics of a nuclear reactor]. Preprint IPM im. V.M. Keldysha RAN-92 - Preprint Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)-92. Moscow, 1996.
5. LAPACK -Linear Algebra PACKage. Available at: http://www.netlib.org/lapack/.(accessed 22.11.2016).
6. International criticality safety benchmark evaluation, nea/nsc/doc(95)03/II.
7. Manturov G.N., Nikolaev M.N., Tsibuly A.M. Sistema gruppovykh konstant BNAB-93. Chast' 1: yadernye konstanty dlya rascheta neytronnykh i fotonnykh poley izlucheniya [BNAB-93 group data library. Part 1: nuclear data for the calculation of neutron and photon radiation fields]. Voprosy atomnoy nauki i tekhniki. Seriya:Yadernye konstanty - Problems of atomic science and technology. Series: Nuclear Constants, 1996, no.1, pp.59-98.
8. Zemskov E.A. K voprosu optimizatsii algoritma vneshnikh i vnutrennikh iteratsiy v diffuzionnom raschete yadernogo reaktora [On the optimization of the algorithm for external and internal iterations in the diffusion calculation of a nuclear reactor]. Preprint FEI-1942 - Preprint IPPE-1942. Obninsk, 1988.
9. Zemskov E.A. K voprosu optimizatsii algoritma vneshnikh i vnutrennikh iteratsiy v diffuzionnom raschete yadernogo reaktora (chast' II) [On the optimization of the algorithm for external and internal iterations in the diffusion calculation of a nuclear reactor (Part II)]. Preprint FEI-2135 - Preprint IPPE-2135. Obninsk, 1990.
10. Arnol'd V.I. Obyknovennye differentsial'nye uravneniya [Ordinary differential equations]. Moscow, Nauka, 1971. 275 p.
11. Zemskov E.A. Analiticheskie testy po kinetike neytronov bystrogo reaktora v odnomernykh geometriyakh (plastina, tsilindr, sfera) [Analytical tests on the kinetics of fast reactor neutrons in one-dimensional geometries (plate, cylinder, sphere)]. Voprosy Atomnoy Nauki i Tekhniki. Seriya: Yaderno-reaktornye konstanty - Problems of Atomic Science and Technology. Series: Nuclear and Reactor Constans, 2016, no. 2, pp. 15-31.
12. Samokhin D.S., Khorosanov G.L., Tormyshev I.V., Zemskov E.A., Gostev A.L., Terekhova A.M., Kuz'michev S.A. Bystryy svintsovyy reaktor maloy moshchnosti dlya uchebnykh tseley [Fast lead-free reactor for educational purposes]. Izvestiya vuzov. Yadernaya energetika - Proseedings of Universities. Nuclear Power Engineering, 2015, no.3, pp. 135-143.
13. Pupko V.Ya. Obshchee rassmotrenie voprosov kinetiki neytronnogo potoka v nerazmnozhayushchikh i mul'tiplitsiruyushchikh sredakh s ispol'zovaniem vysshikh garmonik [General consideration of the problems of the neutron flux kinetics in non-multiplying and multiplying media using higher harmonics]. Trudy vsesoyuznoy letney shkoly fizikov po impul'snym i statisticheskim metodam issledovaniya reaktorov [Proc. All-Union Summer School of Physicists on pulse and statistical methods for investigating reactors]. Obninsk, 1969, vol. 1, pp. 286-361.
14. Godunov S.K. Sovremennye aspekty lineynoy algebry [Modern Aspects of Linear Algebra]. Novosibirsk, Nauchnaya kniga Publ., 1997.