# Uncertainty Quantification Spectral Technique for the Stochastic Point Reactor with Random Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spectral Stochastic and Random Techniques

^{th}–order Hermite polynomial.

## 3. The Point-Reactor Model with Stochastic Variations

^{3}) by using the relation between the thermal/neutronic power $P(t)$ and the average neutron density $n$, where:

## 4. Applications of the Hybrid Technique

^{(1)}and take the expectation with respect to WIE basis. Details of the expectation using WIE basis are described in [1].

## 5. Results and Discussions

^{−1}), $\nu =2.5$, $\Lambda =0.0000179$ (s), $\beta =\left\{0.000215,\text{}0.001424,\text{}0.001274,\text{}0.002568,\text{}0.000748,\text{}0.000273\right\}$. The solution is initiated by assuming external rate $q$ = 10,000 neutrons per second.

^{−1}), $\nu =2.5$, $\Lambda =0.00002$ (s), $q=0.0$, $\beta =\left\{0.000266,\text{}0.001491,\text{}0.001316,\text{}0.002849,\text{}0.000896,\text{}0.000182\right\}$, $T=0.1$ (s). The initial conditions were taken as ${n}_{0}^{(0)}=100$ and ${c}_{k,0}^{(0)}={n}_{0}^{(0)}\text{\hspace{0.17em}}{\beta}_{k}/\left({\lambda}_{k}\Lambda \right);\text{}k=1\cdots 6$. The comparison shows consistency of the results with the literature work, as in Table 2.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- El-Beltagy, M.; Noor, A. Analysis of the stochastic point reactor using Wiener-Hermite expansion. Ann. Nucl. Energy
**2019**, 134, 250–257. [Google Scholar] [CrossRef] - Das, S.; Pan, I.; Das, S. Fractional order fuzzy control of nuclear reactor power with thermal-hydraulic effects in the presence of random network induced delay and sensor noise having long range dependence. Energy Conserv. Manag.
**2013**, 68, 200–218. [Google Scholar] [CrossRef] [Green Version] - Hayes, J.; Allen, E. Stochastic point kinetics equations in nuclear reactor dynamics. Ann. Nucl. Energy
**2005**, 32, 572–587. [Google Scholar] [CrossRef] [Green Version] - Ray, S. Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor. Ann. Nucl. Energy
**2015**, 49, 154–159. [Google Scholar] - Ayyoubzadeh, S.; Vosoughi, N. An alternative stochastic formulation for the point reactor. Ann. Nucl. Energy
**2014**, 63, 691–695. [Google Scholar] [CrossRef] - Skavdahl, I.; Utgikar, V.; Christensen, R.; Chen, M.; Sun, X.; Sabharwall, P. Control of advanced reactor-coupled heat exchanger system: Incorporation of reactor dynamics in system response to load disturbances. Nucl. Eng. Technol.
**2016**, 48, 1349–1359. [Google Scholar] [CrossRef] [Green Version] - Kazeminejad, H. Thermal-hydraulic modeling of flow inversion in a research reactor. Ann. Nucl. Energy
**2008**, 35, 1813–1819. [Google Scholar] [CrossRef] - LeMaître, O.; Knio, O. Spectral Methods for Uncertainty Quantification, with Applications to Computational Fluid Dynamics; Springer: Amsterdam, The Netherlands, 2010. [Google Scholar]
- El-Beltagy, M. A practical comparison between the spectral techniques in solving the SDEs. Eng. Comput.
**2019**, 36, 2369–2402. [Google Scholar] [CrossRef] - El-Beltagy, M.; El-Tawil, M. Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm. Appl. Math. Model.
**2013**, 37, 7174–7192. [Google Scholar] [CrossRef] - Meecham, W. Scaleless algebraic energy spectra for the incompressible Navier–Stokes equation; relation to other nonlinear problems. Mar. Syst.
**1999**, 21, 113–130. [Google Scholar] [CrossRef] - Ghanem, R.; Spanos, P. Stochastic Finite Elements: A Spectral Approach; Springer: New York, NY, USA, 1991. [Google Scholar]
- Xiu, D.; Karniadakis, G. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. Siam J. Sci. Comput.
**2002**, 24, 619–644. [Google Scholar] [CrossRef] - Hamdia, K.; Marino, M.; Zhuang, X.; Wriggers, P.; Rabczuk, T. Sensitivity analysis for the mechanics of tendons and ligaments: Investigation on the effects of collagen structural properties via a multiscale modeling approach. Int. J. Numer. Methods Biomed. Eng.
**2019**, 35, e3209. [Google Scholar] [CrossRef] [PubMed] - Allen, E. Modeling with Itô Stochastic Differential Equations. Mathematical Modelling: Theory and Applications; Springer: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Puchalski, B.; Rutkowski, T.; Duzinkiewicz, K. Nodal models of pressurized water reactor core for control purposes—A comparison study. Nucl. Eng. Des.
**2017**, 322, 444–463. [Google Scholar] [CrossRef] - El-Sefy, M.; Ezzeldin, M.; El-Dakhakhni, W.; Wiebe, L.; Nagasaki, S. System dynamics simulation of the thermal dynamic processes in nuclear power plants. Nucl. Eng. Technol.
**2019**, 51, 1540–1553. [Google Scholar] [CrossRef] - LeMaître, O.; Knio, O. PC analysis of stochastic differential equations driven by Wiener noise. Reliab. Eng. Syst. Saf.
**2015**, 135, 107–124. [Google Scholar] [CrossRef] - Da Silva, M.; Vilhena, M.; Bodmann, B.; Vasques, R. The solution of the neutron point kinetics equation with stochastic extension: An analysis of two moments. In Proceedings of the 2015 International Nuclear Atlantic Conference—INAC 2015, Sao Paulo, Brazil, 4–9 October 2015. [Google Scholar]

**Figure 2.**(

**a**) Deviations in power due to random parameters. (

**b**) Deviations in delayed precursors due to random parameters.

**Figure 3.**(

**a**) Deviations in neutronic power due to noise; (

**b**) deviations in delayed precursors due to noise.

**Figure 4.**(

**a**) Deviations in power due to noise, parameters, and mixed in case of $\alpha $ = 5.0%. (

**b**) Deviations in power due to noise, parameters and mixed in case of $\alpha $ = 50.0%.

Random Deviations $\mathit{\alpha}$% | Power Deviations ${\mathit{\sigma}}_{\mathit{P}}\text{}\%$ | Precursor Deviations ${\mathit{\sigma}}_{\mathit{c}}\text{}\%$ |
---|---|---|

2.50 | 0.8 | 13.1 |

5.00 | 1.5 | 26.2 |

7.50 | 2.3 | 39.3 |

10.0 | 3.1 | 52.4 |

Monte-Carlo [3] | Stochastic PCA [19] | SSPK [5] | Current Work | ||
---|---|---|---|---|---|

$E\left[n(0.1)\right]$ | 183.04 | 187.05 | 184.8 | 179.91 | |

$\sigma \left[n(0.1)\right]$ | 168.79 | 167.83 | 186.96 | 183.5 | |

$E\left[{\displaystyle \sum _{j=1}^{6}{c}_{j}(0.1)}\right]$ | 4.478 × 10^{5} | 4.488 × 10^{5} | 4.489 × 10^{5} | 4.489 × 10^{5} | |

$\sigma \left[{\displaystyle \sum _{j=1}^{6}{c}_{j}(0.1)}\right]$ | 1495.70 | 1475.55 | 982.64 | 1051.24 |

$\mathit{\alpha}$ | Neutronic Power | Noise Std. Dev. | Random Std. Dev. | Mixed Std. Dev. | Total Std. Dev. |
---|---|---|---|---|---|

5 | 601.3 | 848.9 | 0.92 | 0.032 | 848.91 |

10 | 597.9 | 846.5 | 1.84 | 0.064 | 846.51 |

15 | 592.3 | 842.5 | 2.76 | 0.096 | 842.50 |

25 | 574.1 | 829.5 | 4.60 | 0.160 | 829.53 |

50 | 488.1 | 765.2 | 9.21 | 0.280 | 765.26 |

75 | 340.8 | 640.4 | 13.78 | 0.33 | 640.57 |

100 | 56.0 | 285.4 | 14.47 | 0.16 | 285.80 |

**Table 4.**Parameters used in thermal hydraulics for first case [16].

$\mathbf{Nominal}\text{}\mathbf{Power}\text{}{\mathit{P}}_{\mathit{N}}=3436\text{}\left(\mathbf{MW}\right)$ | $\mathbf{Fuel}\text{}\mathbf{Mass}{\mathit{m}}_{\mathit{f}}=101,033\text{}\mathbf{kg}$ |
---|---|

Nominal neutron density ${n}_{N}$ = 249,952,820/cm^{3} | Fuel specific heat ${c}_{pf}$ = 247 J/kg °C |

Mass flow rate of coolant ${W}_{c}$ = 19,852 kg/s | Coolant mass ${m}_{c}$ = 11,196 kg |

Power fraction $f$ = 0.974 | Coolant specific heat ${c}_{pc}$ = 5820 J/kg °C |

Heat transfer coefficient $h$ = 1136 W/m^{2} °C | Effective transfer area $A$ = 5565 m^{2} |

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**MDPI and ACS Style**

Alaskary, S.; El-Beltagy, M.
Uncertainty Quantification Spectral Technique for the Stochastic Point Reactor with Random Parameters. *Energies* **2020**, *13*, 1297.
https://doi.org/10.3390/en13061297

**AMA Style**

Alaskary S, El-Beltagy M.
Uncertainty Quantification Spectral Technique for the Stochastic Point Reactor with Random Parameters. *Energies*. 2020; 13(6):1297.
https://doi.org/10.3390/en13061297

**Chicago/Turabian Style**

Alaskary, Safa, and Mohamed El-Beltagy.
2020. "Uncertainty Quantification Spectral Technique for the Stochastic Point Reactor with Random Parameters" *Energies* 13, no. 6: 1297.
https://doi.org/10.3390/en13061297