Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain
In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fa...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Brazilian Radiation Protection Society (Sociedade Brasileira de Proteção Radiológica, SBPR)
2019-06-01
|
| Series: | Brazilian Journal of Radiation Sciences |
| Subjects: | |
| Online Access: | https://bjrs.org.br/revista/index.php/REVISTA/article/view/389 |
| _version_ | 1798030458876854272 |
|---|---|
| author | Fernanda Tumelero Celso M. F. Lapa Bardo E. J Bodmann Marco T. Vilhena |
| author_facet | Fernanda Tumelero Celso M. F. Lapa Bardo E. J Bodmann Marco T. Vilhena |
| author_sort | Fernanda Tumelero |
| collection | DOAJ |
| description | In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution. |
| first_indexed | 2024-04-11T19:41:43Z |
| format | Article |
| id | doaj.art-7103fde91e724f71866c0d0672689852 |
| institution | Directory Open Access Journal |
| issn | 2319-0612 |
| language | English |
| last_indexed | 2024-04-11T19:41:43Z |
| publishDate | 2019-06-01 |
| publisher | Brazilian Radiation Protection Society (Sociedade Brasileira de Proteção Radiológica, SBPR) |
| record_format | Article |
| series | Brazilian Journal of Radiation Sciences |
| spelling | doaj.art-7103fde91e724f71866c0d06726898522022-12-22T04:06:42ZengBrazilian Radiation Protection Society (Sociedade Brasileira de Proteção Radiológica, SBPR)Brazilian Journal of Radiation Sciences2319-06122019-06-0172B10.15392/bjrs.v7i2B.389Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domainFernanda Tumelero0Celso M. F. Lapa1Bardo E. J Bodmann2Marco T. Vilhena3Programa de Pós-Graduação em Engenharia Mecânica (PROMEC) Universidade Federal do Rio Grande do Sul (UFRGS)Instituto de Engenharia Nuclear (IEN)/CNENPrograma de Pós-Graduação em Engenharia Mecânica (PROMEC) Universidade Federal do Rio Grande do Sul (UFRGS)Programa de Pós-Graduação em Engenharia Mecânica (PROMEC) Universidade Federal do Rio Grande do Sul (UFRGS)In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution.https://bjrs.org.br/revista/index.php/REVISTA/article/view/389neutron diffusion equationTaylor seriesmodified Adomian Decomposition method |
| spellingShingle | Fernanda Tumelero Celso M. F. Lapa Bardo E. J Bodmann Marco T. Vilhena Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain Brazilian Journal of Radiation Sciences neutron diffusion equation Taylor series modified Adomian Decomposition method |
| title | Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain |
| title_full | Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain |
| title_fullStr | Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain |
| title_full_unstemmed | Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain |
| title_short | Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain |
| title_sort | analytical representation of the solution of the space kinetic diffusion equation in a one dimensional and homogeneous domain |
| topic | neutron diffusion equation Taylor series modified Adomian Decomposition method |
| url | https://bjrs.org.br/revista/index.php/REVISTA/article/view/389 |
| work_keys_str_mv | AT fernandatumelero analyticalrepresentationofthesolutionofthespacekineticdiffusionequationinaonedimensionalandhomogeneousdomain AT celsomflapa analyticalrepresentationofthesolutionofthespacekineticdiffusionequationinaonedimensionalandhomogeneousdomain AT bardoejbodmann analyticalrepresentationofthesolutionofthespacekineticdiffusionequationinaonedimensionalandhomogeneousdomain AT marcotvilhena analyticalrepresentationofthesolutionofthespacekineticdiffusionequationinaonedimensionalandhomogeneousdomain |