# A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models

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## Abstract

**:**

## 1. Introduction

## 2. The Stochastic System Setup

^{th}change in $\mathsf{\Delta}t$ is set to be ${p}_{j}\mathsf{\Delta}t$; then the average change in $X$ in $\mathsf{\Delta}t$ is

^{th}possible change in $\mathsf{\Delta}t$. Moreover, the covariance matrix of the random change in $\mathsf{\Delta}t$ is computed as

^{th}precursor density, ${\beta}_{i}$ is the average of the i

^{th}neutron precursor fraction, $\beta $ is the average of the total neutron precursor fraction, ${\lambda}_{i}$ is the decayed constant of the i

^{th}neutron precursor, $\alpha $ is defined as $1/\nu $, where $\nu $ is the total number of neutron released per fission, ${\mathsf{\Sigma}}_{a}$ is the absorption cross-section, and $q$ is the average number of neutron injections per unit volume.

^{th}random change in $\mathsf{\Delta}t$, which could occur in $\mathsf{\Delta}t$ with probability ${p}_{j}\mathsf{\Delta}t$ defined with an error of order $O\left({\left(\mathsf{\Delta}t\right)}^{2}\right)$. Hence, the expectation and variance of the i

^{th}component and j

^{th}random change in $\mathsf{\Delta}t$ are given by ${\lambda}_{j,i}{p}_{j}\mathsf{\Delta}t$ and ${\lambda}_{j,i}^{2}{p}_{j}\Delta t$, respectively. In the following sections, we review two modeling procedures to build stochastic differential equation models, which approximate the process in Equation (7), along with its application to the point kinetic equations.

## 3. The First Modeling Procedure and the Stochastic Point Kinetic Equations (SPK)

#### 3.1. The First Procedure

#### 3.2. The Stochastic Point Kinetic Model

## 4. Alternative Modeling Procedure and the Langevin Point Kinetic Equations (LPK)

#### 4.1. The Second Procedure

^{th}column defined as

#### 4.2. Equivalence of the Two Procedures

#### 4.3. The Langevin Point Kinetic Model (LPK)

#### 4.4. The Simplified Stochastic Point Kinetic Model (SSPK)

## 5. Euler Method

## 6. Test Cases and Results

#### 6.1. First Case

#### 6.2. Second Case

#### 6.3. Third Case

#### 6.4. Fourth Case

#### 6.5. Fifth Case

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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MC | SPK | LPK | SSPK | |
---|---|---|---|---|

$\mathit{E}\left(\mathit{n}\left(\mathbf{2}\right)\right)$ | 400.03 | 398.42 | 398.60 | 398.19 |

$\mathit{\sigma}\left(\mathit{n}\left(\mathbf{2}\right)\right)$ | 27.311 | 31.072 | 31.166 | 31.511 |

$\mathit{E}\left({\mathit{c}}_{\mathbf{1}}\left(\mathbf{2}\right)\right)$ | 300.00 | 300.13 | 300.15 | 299.97 |

$\mathit{\sigma}\left({\mathit{c}}_{\mathbf{1}}\left(\mathbf{2}\right)\right)$ | 7.8073 | 8.0586 | 8.0769 | 8.0410 |

**MC**-Monte Carlo;

**SPK**-Stochastic Point Kinetic;

**LPK**- Langevin point kinetic;

**SSPK**- Simplified Stochastic Point Kinetic.

SPK | LPK | SSPK | |
---|---|---|---|

Time (s) | 161.80 | 69.367 | 64.096 |

MC | SPK | LPK | SSPK | |
---|---|---|---|---|

$\mathit{E}\left(\mathit{n}\left(\mathbf{0}\mathbf{.}\mathbf{1}\right)\right)$ | 183.04 | 187.41–1.516 × 10^{−11}i | 185.18 | 186.60 |

$\mathit{\sigma}\left(\mathit{n}\left(\mathbf{0}\mathbf{.}\mathbf{1}\right)\right)$ | 168.79 | 166.37 + 5.258 × 10^{−11}i | 161.52 | 165.67 |

$\mathit{E}\left(\mathbf{\sum}{\mathit{c}}_{\mathit{i}}\left(\mathbf{0}\mathbf{.}\mathbf{1}\right)\right)$ | 4.478 × 10^{5} | 4.490 × 10^{5}–3.084 × 10^{−9}i | 4.491 × 10^{5} | 4.490 × 10^{5} |

$\mathit{\sigma}\left(\mathbf{\sum}{\mathit{c}}_{\mathit{i}}\left(\mathbf{0}\mathit{.}\mathit{1}\right)\right)$ | 1495.7 | 1933.6–5.397 × 10^{−10}i | 1952.7 | 1899.0 |

SPK | LPK | SSPK | |
---|---|---|---|

Time (s) | 481.51 | 250.05 | 244.37 |

MC | SPK | LPK | SSPK | |
---|---|---|---|---|

$\mathit{E}\left(\mathit{n}\left(\mathbf{0}\mathbf{.}\mathbf{001}\right)\right)$ | 135.67 | 135.63–1.334 × 10^{−12}i | 133.94 | 135.17 |

$\mathit{\sigma}\left(\mathit{n}\left(\mathbf{0}\mathbf{.}\mathbf{001}\right)\right)$ | 93.376 | 92.737 + 2.196 × 10^{−12}i | 93.638 | 92.853 |

$\mathit{E}\left(\mathbf{\sum}{\mathit{c}}_{\mathit{i}}\left(\mathbf{0}\mathbf{.}\mathbf{001}\right)\right)$ | 4.464 × 10^{5} | 4.464 × 10^{5} + 8.916 × 10^{−12}i | 4.464 × 10^{5} | 4.464 × 10^{5} |

$\mathit{\sigma}\left(\mathbf{\sum}{\mathit{c}}_{\mathit{i}}\left(\mathbf{0}\mathbf{.}\mathbf{001}\right)\right)$ | 16.226 | 19.408–1.374 × 10^{−11}i | 19.495 | 19.163 |

SPK | LPK | SSPK | |
---|---|---|---|

Time (s) | 18.478 | 9.9243 | 9.7573 |

MC | SPK | LPK | SSPK | |
---|---|---|---|---|

$\mathit{E}\left(\mathit{t}\right)$ | 33.136 | 33.098 | 33.098 | 33.115 |

$\mathit{\sigma}\left(\mathit{t}\right)$ | 2.0886 | 2.7372 | 2.7061 | 2.6616 |

SPK | LPK | SSPK | |

Time (s) | 1056.7 | 484.83 | 441.48 |

MC | SPK | LPK | SSPK | |
---|---|---|---|---|

$\mathit{E}\left(\mathit{t}\right)$ | 31.8 | 30.479 | 30.416 | 30.332 |

$\mathit{\sigma}\left(\mathit{t}\right)$ | 4.5826 | 5.1888 | 5.0252 | 4.9750 |

SPK | LPK | SSPK | |

Time (s) | 1736.4 | 740.86 | 714.70 |

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**MDPI and ACS Style**

Elsayed, A.; El-Beltagy, M.; Al-Juhani, A.; Al-Qahtani, S.
A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models. *Energies* **2021**, *14*, 955.
https://doi.org/10.3390/en14040955

**AMA Style**

Elsayed A, El-Beltagy M, Al-Juhani A, Al-Qahtani S.
A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models. *Energies*. 2021; 14(4):955.
https://doi.org/10.3390/en14040955

**Chicago/Turabian Style**

Elsayed, Alamir, Mohamed El-Beltagy, Amnah Al-Juhani, and Shorooq Al-Qahtani.
2021. "A New Model for the Stochastic Point Reactor: Development and Comparison with Available Models" *Energies* 14, no. 4: 955.
https://doi.org/10.3390/en14040955