Elshin A.V.
The Institute of Nuclear Power Engineering, “St. Petersburg Polytechnic University named after Peter the Great”, Sosnoby Bor, Russia
Currently there are quite a few methods for “straightforward” solution of neutron transport equation: the Monte Carlo method, discrete-ordinates method, the method of characteristics and others. The problem of the solution consists in the fact that for the real systems the dimension of the problem turns out to be tremendous, and the solution cannot be obtained promptly. The method of solution that is presently the most widely used is the homogenization method. This method reduces a high-dimension problem to a few problems of smaller dimensions. The neutron transport equation is solved only within the framework of a few (of different types) elementary cells of the reactor. These neutron distributions are used for the cell homogenization, i.e. a heterogeneous cell is replaced with abstract homogeneous medium “with the same multiplying and diffusion characteristics”, and the neutron distribution in the reactor on the whole is sought as the solution of a few-group diffusion equation in the system with piecewise-constant properties. The homogenization method is justified and specified by means of the surface harmonic method proposed in the last century, which in some way formalized the list of the required neutron distributions in elementary cells and the formulas to calculate coefficients of finite-difference few-group equations needed to calculate the reactor neutron distribution on the whole. This paper demonstrates that in the surface harmonic method the transfer to higher approximations (doing without diffusion approximation) is only carried out by increasing the dimension of equation matrices-coefficients (and thus, increasing the number of testing functions used to describe neutron distribution in elementary cells of the heterogeneous reactor).
1. Kondrushin A.E. Razvitie metoda poverhnostnykh garmonik dlya resheniya zadach nejtronnoj prostranstvennoj kinetiki v yadernykh reaktorakh. Diss. kand. fiz.-mat. nauk [Development of Surface Harmonic Method to Solve the Problems of Space Neutron Kinetics in Nuclear Reactors. Cand. phys. math. sci. diss.]. Moscow, 2013.
2. Laletin N.I. Basic Principles for Developing Equations for Heterogeneous Reactors – A Modification of the Homogenization Method. Nuclear Science and Engineering, 1983, Vol. 85, pp. 133-138.
3. Laletin N.I., El'shin A.V. Sistema utochnennykh konechno-raznostnykh uravnenij dlya trekhmernogo geterogennogo reaktora [The System of Specified Finite-Difference Equations for 3D Heterogeneous Reactor]. Atomnaya energiya - Atomic Energy. 1986, Vol. 60, 2, pp. 96-99.
4. Boyarinov V.F. Trekhmernye uravneniya geterogennogo reaktora v metode poverkhnostnykh garmonik s odnoj neizvestnoj na yachejko-gruppu [3D Heterogeneous Reactor Equations in the Surface Harmonic Method with one Unknown per Cell-Group]. Atomnaya energiya - Atomic Energy. 1992, Vol. 72, 3, pp. 227-231.
5. Laletin N.I., Kovalishin A.A. The Influence of the Higher Surfase Harmonics Method by Calculations RBMK and VVER Lattices. Proceeding of International Conference PHYSOR-96, Mito, Japan, 1996, Vol. 1, A-249.
6. Laletin N.I., El'shin A.V. Vyvod konechno-raznostnykh uravnenij geterogennogo reaktora 2. Kvadratnaya, treugol'naya i dvojnaya reshetka blokov [Derivation of Finite-Difference Equations for the Heterogeneous Reactor 2. Square, Triangle and Double Lattice]. Preprint IAE 3458/5 - Preprint IAE 3458/5. Moscow, 1981.
7. El'shin A.V., Abdullaev A.M. O sootnosheniyakh vzaimnosti i vysokikh priblizheniyakh metoda poverkhnostnykh garmonik [On Reciprocity Relations and High Approximations of the Surface Harmonic Method]. Trudy mezhvedomstvennogo 23 seminara «Nejtronno-fizicheskie problemy atomnoj energetiki s zamknutym toplivnym ciklom» (Nejtronika-2012) [Proc. 23th Interagency Workshop “Neutronic Problems in Nuclear Power with Closed Fuel Cycle” (Neutronics-2012)]. Obninsk, 2013, 2, pp. 515-521.
8. El'shin A.V. Poluchenie konechno-raznostnykh uravnenij dlya cennosti nejtronov v geterogennom reaktore metodom poverkhnostnykh garmonik [Derivation of Finite-Difference Equations for Neutron Importance in the Heterogeneous Reactor with the Surfase Harmonics Method]. Atomnaya energiya - Atomic Energy. 2005, Vol. 98, 5, pp. 323-332.
9. El'shin A.V. Poluchenie konechno-raznostnykh uravnenij geterogennogo reaktora s prostranstvennoj kinetikoj [Derivation of Finite-Difference Equations for the Heterogeneous Reactor with Space Kinetics]. Atomnaya energiya - Atomic Energy. 2007, Vol. 103, 4, pp.222-232.
