### Abstract

This work focuses on the critical value calculations and the solving of the multi-group neutron diffusion equation, which is frequently introduced in a senior level reactor physics textbook. keff denotes the effective neutron multiplication factor (Ref. 1, Duderstadt). The mathematical structure of solutions and the review of values of the solutions have attained the status of a mature academic subtopic. On the other hand, for the cases of multi-group neutrons, the conventionally known texts of neutron diffusion have not fully succeeded to show the formulations and the vital connections to the relevant numerical calculations in order to directly follow through (without interactive decision steps to proceed) to concisely find the values of keff and/or of critical size, critical enrichment value, etc. In consideration of 2-group diffusion for cylindrically shaped and analogously shaped cores and with neglect of realistic boundary conditions of n-leakage from surface, one can readily find two eigensolutions and with two distinct eigenvalues for buckling in order to satisfy the given values of keff and of the critically sensitive physical parameters such as enrichment. However, if the boundary conditions for the highest energy group and the lowest energy group are not with identical values for the corresponding diffusion lengths, then it becomes significantly difficult to adjust parameters such as enrichment in order to get the buckling quantities of the two eigen-solutions along with the amplitudes selected so as to get the first function of the eigenmode and the second function of the eigenmode of the 2-group system to satisfy all of the boundary conditions required of the reactor core. For a spherical core governed by 2-group neutrons, the scalar flux has the spatial form of sin(B{_{1}} r)/r + C{_{2}}sinh(B{^{2{}} r)/r, where B{1} 2 and B{2} 2 are the two possible values for buckling (Ref. 2). If this were a one-group neutron reactor, scalar flux would simply equal A{0} sin('BB[o]' r)/r., and it is simple enough to determine what 'BB[o]' equals. However, the expression for 2-group flux has the burden of having B{_{1}}, B{_{2}}, and C{_{2}} to satisfy extrapolation or the Robin boundary conditions (Ref. 3) for neutrons at energy-1 as well as at energy-2. This makes the challenge of determining the contingent relationship of keff with Enrichment Ratio within two-group diffusion more than five times more intensive numerically than it is for onegroup analysis. This paper offers an insightful method to find keff or alternatively the value of one of the vital critical parameters which either directly or implicitly effects keff. The finite element inspired method used to do this involves the construction of a matrix which has to be solved for keff and a matching critical parameter-such as enrichment or diameter (Ref. 4). A sequence of these matrices can be created: Typically the sequence consists of a 4 by 4 matrix, 6 by 6 matrix, etc. The solutions of the determinants of the matrices in this sequence give us the lowest eigenvalues of either keff or of the first critical parameter. It has been verified in our worked examples and is shown in this paper that this sequence of values of keff or of critical parameter converges to a constant value as the order of the matrix increases. Often, progression to the 12 by 12 matrix is sufficient in order to find the convergent value for cylindrical, spherical, and rectilinear cores.

Original language | English (US) |
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Title of host publication | Proceedings of the 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018 |

Publisher | American Nuclear Society |

Pages | 721-731 |

Number of pages | 11 |

ISBN (Electronic) | 9780894487552 |

State | Published - Jan 1 2018 |

Externally published | Yes |

Event | 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018 - Charlotte, United States Duration: Apr 8 2018 → Apr 11 2018 |

### Other

Other | 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018 |
---|---|

Country | United States |

City | Charlotte |

Period | 4/8/18 → 4/11/18 |

### Fingerprint

### ASJC Scopus subject areas

- Energy Engineering and Power Technology
- Nuclear Energy and Engineering

### Cite this

*Proceedings of the 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018*(pp. 721-731). American Nuclear Society.

**A semi-Analytical matrix to extract keff and/or critical parameters of fissile cores -as an aid for deterministic calculations of multi-group neutronics.** / Steinfelds, Eric V.; Ganapol, Barry D; Andrew, Keith.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018.*American Nuclear Society, pp. 721-731, 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018, Charlotte, United States, 4/8/18.

}

TY - GEN

T1 - A semi-Analytical matrix to extract keff and/or critical parameters of fissile cores -as an aid for deterministic calculations of multi-group neutronics

AU - Steinfelds, Eric V.

AU - Ganapol, Barry D

AU - Andrew, Keith

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This work focuses on the critical value calculations and the solving of the multi-group neutron diffusion equation, which is frequently introduced in a senior level reactor physics textbook. keff denotes the effective neutron multiplication factor (Ref. 1, Duderstadt). The mathematical structure of solutions and the review of values of the solutions have attained the status of a mature academic subtopic. On the other hand, for the cases of multi-group neutrons, the conventionally known texts of neutron diffusion have not fully succeeded to show the formulations and the vital connections to the relevant numerical calculations in order to directly follow through (without interactive decision steps to proceed) to concisely find the values of keff and/or of critical size, critical enrichment value, etc. In consideration of 2-group diffusion for cylindrically shaped and analogously shaped cores and with neglect of realistic boundary conditions of n-leakage from surface, one can readily find two eigensolutions and with two distinct eigenvalues for buckling in order to satisfy the given values of keff and of the critically sensitive physical parameters such as enrichment. However, if the boundary conditions for the highest energy group and the lowest energy group are not with identical values for the corresponding diffusion lengths, then it becomes significantly difficult to adjust parameters such as enrichment in order to get the buckling quantities of the two eigen-solutions along with the amplitudes selected so as to get the first function of the eigenmode and the second function of the eigenmode of the 2-group system to satisfy all of the boundary conditions required of the reactor core. For a spherical core governed by 2-group neutrons, the scalar flux has the spatial form of sin(B{1} r)/r + C{2}sinh(B{2{} r)/r, where B{1} 2 and B{2} 2 are the two possible values for buckling (Ref. 2). If this were a one-group neutron reactor, scalar flux would simply equal A{0} sin('BB[o]' r)/r., and it is simple enough to determine what 'BB[o]' equals. However, the expression for 2-group flux has the burden of having B{1}, B{2}, and C{2} to satisfy extrapolation or the Robin boundary conditions (Ref. 3) for neutrons at energy-1 as well as at energy-2. This makes the challenge of determining the contingent relationship of keff with Enrichment Ratio within two-group diffusion more than five times more intensive numerically than it is for onegroup analysis. This paper offers an insightful method to find keff or alternatively the value of one of the vital critical parameters which either directly or implicitly effects keff. The finite element inspired method used to do this involves the construction of a matrix which has to be solved for keff and a matching critical parameter-such as enrichment or diameter (Ref. 4). A sequence of these matrices can be created: Typically the sequence consists of a 4 by 4 matrix, 6 by 6 matrix, etc. The solutions of the determinants of the matrices in this sequence give us the lowest eigenvalues of either keff or of the first critical parameter. It has been verified in our worked examples and is shown in this paper that this sequence of values of keff or of critical parameter converges to a constant value as the order of the matrix increases. Often, progression to the 12 by 12 matrix is sufficient in order to find the convergent value for cylindrical, spherical, and rectilinear cores.

AB - This work focuses on the critical value calculations and the solving of the multi-group neutron diffusion equation, which is frequently introduced in a senior level reactor physics textbook. keff denotes the effective neutron multiplication factor (Ref. 1, Duderstadt). The mathematical structure of solutions and the review of values of the solutions have attained the status of a mature academic subtopic. On the other hand, for the cases of multi-group neutrons, the conventionally known texts of neutron diffusion have not fully succeeded to show the formulations and the vital connections to the relevant numerical calculations in order to directly follow through (without interactive decision steps to proceed) to concisely find the values of keff and/or of critical size, critical enrichment value, etc. In consideration of 2-group diffusion for cylindrically shaped and analogously shaped cores and with neglect of realistic boundary conditions of n-leakage from surface, one can readily find two eigensolutions and with two distinct eigenvalues for buckling in order to satisfy the given values of keff and of the critically sensitive physical parameters such as enrichment. However, if the boundary conditions for the highest energy group and the lowest energy group are not with identical values for the corresponding diffusion lengths, then it becomes significantly difficult to adjust parameters such as enrichment in order to get the buckling quantities of the two eigen-solutions along with the amplitudes selected so as to get the first function of the eigenmode and the second function of the eigenmode of the 2-group system to satisfy all of the boundary conditions required of the reactor core. For a spherical core governed by 2-group neutrons, the scalar flux has the spatial form of sin(B{1} r)/r + C{2}sinh(B{2{} r)/r, where B{1} 2 and B{2} 2 are the two possible values for buckling (Ref. 2). If this were a one-group neutron reactor, scalar flux would simply equal A{0} sin('BB[o]' r)/r., and it is simple enough to determine what 'BB[o]' equals. However, the expression for 2-group flux has the burden of having B{1}, B{2}, and C{2} to satisfy extrapolation or the Robin boundary conditions (Ref. 3) for neutrons at energy-1 as well as at energy-2. This makes the challenge of determining the contingent relationship of keff with Enrichment Ratio within two-group diffusion more than five times more intensive numerically than it is for onegroup analysis. This paper offers an insightful method to find keff or alternatively the value of one of the vital critical parameters which either directly or implicitly effects keff. The finite element inspired method used to do this involves the construction of a matrix which has to be solved for keff and a matching critical parameter-such as enrichment or diameter (Ref. 4). A sequence of these matrices can be created: Typically the sequence consists of a 4 by 4 matrix, 6 by 6 matrix, etc. The solutions of the determinants of the matrices in this sequence give us the lowest eigenvalues of either keff or of the first critical parameter. It has been verified in our worked examples and is shown in this paper that this sequence of values of keff or of critical parameter converges to a constant value as the order of the matrix increases. Often, progression to the 12 by 12 matrix is sufficient in order to find the convergent value for cylindrical, spherical, and rectilinear cores.

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M3 - Conference contribution

AN - SCOPUS:85050107330

SP - 721

EP - 731

BT - Proceedings of the 2018 International Congress on Advances in Nuclear Power Plants, ICAPP 2018

PB - American Nuclear Society

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