Authors & Affiliations
Elshin A.V.1,2, Agalina P.V.2
1 Alexandrov Research Institute of Technoloqy, Sosnovy Bor, Russia
2 Institute of Nuclear Power Engineering (branch) of Peter the Great Saint-Petersburg Polytechnic University, Sosnovy Bor, Russia
Elshin A.V. – Senior Researcher, Head of Department, Dr. Sci. (Tech.), Aleksandrov Science and Research Institute of Technology, Institute of Nuclear Power Engineering (branch) of Peter the Great Saint-Petersburg Polytechnic University. Contacts: 72, Koporskoe shosse, Sosnovy Bor, Leningrad region, Russia, 188540. Tel.: +7 (921) 438-48-55; e-mail:
Agalina P.V. – student, Institute of Nuclear Power Engineering (branch) of Peter the Great Saint-Petersburg Polytechnic University.
The paper addresses the application of the surface harmonics method to building a system of finite difference equations for describing the neutron field of a 3D heterogeneous reactor. The reactor is comprised of cells shaped as right-angled parallelepipeds with non-symmetric contents. Neutrons distribution in the cells is presented as a linear combination of trial functions that satisfy the neutron transport equation and some non-uniform boundary conditions allowing to consider the influence of the cell environment on the neutron distribution within the cell. Even given non-symmetric cells, we can build a system of equations (by sewing the neutron distribution in adjacent cells on joint faces), which, if diffusion approximation is used on the cell faces, are similar in appearance to the finite difference approximation of a group-diffusion equation (using a different method to obtain coefficients of equations, with complete matrices of group neutron diffusion coefficients). Finite-difference equations can also be defined by considering the additional neutron distribution components in cells (trial functions), which are necessary for valid transition to particular cases: two-dimensional and one-dimensional geometry. Due to cell asymmetry, additional components are introduced in equations and formulas for equation coefficients are corrected (correction is similar to the introduction of "discontinuity coefficients" used in foreign codes). As it is shown, the system of equations allows us to avoid using the diffusion approximation in reactor calculations since we can now sew together not only the neutron current density and neutron flux but also higher angular momenta at boundaries of cells without changing the form of equations (increasing the dimension of coefficient matrices and dimension of required vectors). Numerical results are presented on an example of 1D single-velocity test problems.
surface harmonics method, 3D heterogeneous reactor model, finite difference equations, non-symmetric unit cells, avoidance of diffusion approximation
1. Laletin N.I., Elshin А.В. Utochnenie metoda gomogenizacii geterjgennogo reactor [Refinement of the method of homogenization of a heterogeneous reactor]. Atomnaya energiya - Atomic Energy, 1977, vol. 43, no. 4, pp. 247–253.
2. Laletin N.I. Basic Principles for Developing Equations for Heterogeneous Reactors – A Modification of the Homogeneous Method. Nuclear Science and Engineering, 1983, vol. 85, pp. 133—138.
3. Kovalishin А.А., Krayushkin А.V., Laletin N.I. et al. Primenenie metoda poverhnostnih garmonic v progranne STEPAN [Application of the surface harmonics method in the STEPAN program]. Atomnaya energiya – Atomic Energy, 2016, vol. 120, no. 2, pp. 249–254.
4. Boyarinov V.F., Kondrushin A.E., Fomichenko P.A. Surface harmonics method for two-dimensional time-dependent neutron transport problems of square-lattice nuclear reactors. Proc. Int. Conf. on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2013). Sun Valley, Idaho, 2013.
5. Elshin A. Poluchenie konechno-raznoctnih uravneniy dlya cennocti neytronov d geterogennom reaktore metodom poverhnostnih garmonik [Obtaining finite-difference equations for the value of neutrons in a heterogeneous reactor by the method of surface harmonics]. Atomnaya energiya – Atomic Energy, 2005, vol. 98, no. 5, pp. 323–332.
6. Elshin A., Konechno-raznoctnie uravneniya dlya raspredeleniya neytronov i ih cennosti v trehmernom geterogennom reaktore c nestrukturirovannoy setkoy [Finite Difference Equations for Neutron Flux and Importance Distribution in 3D Heterogeneous Reactor with Unstructured Mesh]. Voprosy atomnoy nauki i tekhniki. Seriya: Fizika yadernykh reaktorov – Problems of atomic science and technology. Series: Physics of Nuclear Reactors, 2016, no. 5, pp. 36–44.
7. Elshin A., Tenischeva N. Vacuum boundary conditions in surface harmonics method [Vacuum boundary conditions in the method of surface harmonics]. Voprosy Atomnoy Nauki i Tekhniki. Seriya: Yaderno-reaktornye konstanty – Problems of Atomic Science and Technology. Series: Nuclear and Reactor Constans, 2017, no. 1, pp. 41–47.
8. Smith K.S. Spatial Homogenization Methods for Light Water Reactor Analysis. Boston, Massachusetts Inst. Technol., 1980.
9. Trahan T.J., Larsen E.W. An Asymptotic Homogenized Neutron Diffusion Approximation. I Theory/ II Numerical Comparison. Proceeding of Physor2012 – Advances in Reactor Physics – Linking Research, Industry, and Education. Knoxville, Tennessee, USA, 2012.
10. Elshin A., Pakhomov V., Riuchin V. The choice of core unit cells boundaries in surface harmonics method by the test tasks solving example. Journal of Physics: Conference Series, 2017, vol. 784.