# On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant

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## Abstract

**:**

## 1. Introduction

## 2. Lumped Parameter Model for Simulation

## 3. Enhacement of Numerical Efficiency by Singularly Perturbed Point Kinetics

#### 3.1. Background of Singular Perturbation

**Remark**

**1.**

**Remark**

**2.**

- The functions f and g in Equations (12) and (13), respectively, and their first partial derivatives with respect to $(x,z,\epsilon )$ and the first partial derivative of g with respect to t are continuous.
- Initial conditions $\xi (\epsilon )$ and $\eta (\epsilon )$ in Equations (12) and (13), respectively, are smooth functions of $\epsilon $.
- The function $h(t,x)$ in Equation (15) and the Jacobian $[\partial g(t,x,z,0)/\partial z]$ have continuous first partial derivatives with respect to their arguments.
- The reduced-order system in Equation (16) has a unique solution $\tilde{x}(t)$ for $t\in [{t}_{0},{t}_{1}]$ within a compact subset of the solution space.
- The origin in the state space of Equation (19) is an exponentially stable equilibrium of the boundary-layer system.

#### 3.2. Singularly Perturbed Neutron Point Kinetics

**Remark**

**3.**

**Remark**

**4.**

#### 3.3. Example: Sinusoidally Oscillating Reactivity Insertion

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

n | neutron density |

${c}_{i}$ | i-th delayed neutron precursor concentration |

${N}_{c}$ | number of delayed concentration groups ($1\le {N}_{c}\le 6$) |

${\lambda}_{i}$ | effective precursor decay constant for group i |

$\mathsf{\Lambda}$ | effective prompt neutron lifetime |

$\epsilon (t)$ | time-dependent singular perturbation parameter |

$\overline{\epsilon}$ | time-averaged singular perturbation parameter |

$\beta $ | total delayed neutron fraction $\left(\beta \triangleq {\sum}_{i=1}^{{N}_{c}}{\beta}_{i}\right)$ |

$\rho $ | reactivity |

${\rho}_{r}$ | control rod reactivity |

${P}_{c}$ | power transferred from fuel to coolant |

${P}_{e}$ | power removed from the coolant |

${P}_{a}$ | reactor power |

$\mathsf{\Omega}$ | heat transfer coefficient between fuel and coolant |

M | mass flow rate times heat capacity of coolant water |

${T}_{f}$ | average fuel temperature in the reactor |

${T}_{f,r}$ | relative average fuel temperature ($\frac{{T}_{f}}{{T}_{f}(0)})$ |

${T}_{l}$ | coolant temperature at reactor exit |

${T}_{l,r}$ | relative coolant temperature at reactor exit ($\frac{{T}_{l}}{{T}_{l}(0)}$) |

${T}_{e}$ | coolant temperature at reactor entrance |

${T}_{c}$ | average coolant temperature in the reactor |

${f}_{f}$ | fraction of reactor power deposited in the fuel |

${T}_{e0}$ | reference coolant temperature at reactor entrance |

${T}_{c0}$ | reference average coolant temperature |

${\mu}_{f}$ | total heat capacity of the fuel and structural material |

${\mu}_{c}$ | total heat capacity of the reactor coolant |

${\beta}_{i}$ | fraction of neutrons that come from delayed group i |

${\alpha}_{c}$ | coolant temperature coefficient |

${\alpha}_{f}$ | fuel temperature coefficient |

${n}_{r}$ | relative neutron density |

${c}_{r,i}$ | i-th delayed neutron precursor’s relative concentration |

${t}_{0}$ | initial time of transients |

${t}_{1}$ | end time of transients |

## References

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**Figure 1.**Profiles of relative neutron density ${n}_{r}$, delayed neutron precursor concentration ${c}_{r,i},i=1,\cdots ,6$, relative average fuel temperature ${T}_{f,r}$ and relative outlet coolant temperature ${T}_{l,r}$ with integration step size of 1 ms and 0.9 million steps.

**Figure 3.**Profiles of relative neutron density ${n}_{r}$ within the boundary layer, where the solid line

**—–**indicates the profile governed by Equation (4); the dashed line

**- - -**indicates the profile governed by Equation (22); and the dotted line

**$\cdots \cdots $**indicates the profile governed by Equation (23).

**Table 1.**Parameters and Values of parameters used in the simulation [2].

Parameters | Values [Units] |
---|---|

$\mathsf{\Omega}$ | 6.53 (MW/${}^{\circ}$K) |

M | 92.8 (MW/${}^{\circ}$K) |

${\mu}_{f}$ | 26.3 (MW·s/${}^{\circ}$K) |

${\mu}_{c}$ | 70.5 (MW·s/${}^{\circ}$K) |

${T}_{e}$ | 563.15 (${}^{\circ}$K) |

${f}_{f}$ | 0.98 |

${\alpha}_{c}$ | 0.00001 |

${\alpha}_{f}$ | −0.00005 |

${\lambda}_{1}$ | 0.0124 (s${}^{-1}$) |

${\lambda}_{2}$ | 0.0305 (s${}^{-1}$) |

${\lambda}_{3}$ | 0.1110 (s${}^{-1}$) |

${\lambda}_{4}$ | 0.3010 (s${}^{-1}$) |

${\lambda}_{5}$ | 1.1400 (s${}^{-1}$) |

${\lambda}_{6}$ | 3.0100 (s${}^{-1}$) |

$\mathsf{\Lambda}$ | 0.0001 (s) |

${\beta}_{1}$ | 0.000215 |

${\beta}_{2}$ | 0.001424 |

${\beta}_{3}$ | 0.001274 |

${\beta}_{4}$ | 0.002568 |

${\beta}_{5}$ | 0.000748 |

${\beta}_{6}$ | 0.000273 |

$\beta $ | 0.006502 |

${N}_{c}$ | 6 |

${n}_{r}(0)$ | 1 |

${c}_{r,i}(0)$ | 1 |

${T}_{l}(0)$ | 590.09 (${}^{\circ}$K) |

${T}_{f}(0)$ | 951.81 (${}^{\circ}$K) |

${P}_{a}(0)$ | 2500 (MW) |

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**MDPI and ACS Style**

Chen, X.; Ray, A. On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant. *Sci* **2020**, *2*, 36.
https://doi.org/10.3390/sci2020036

**AMA Style**

Chen X, Ray A. On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant. *Sci*. 2020; 2(2):36.
https://doi.org/10.3390/sci2020036

**Chicago/Turabian Style**

Chen, Xiangyi, and Asok Ray. 2020. "On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant" *Sci* 2, no. 2: 36.
https://doi.org/10.3390/sci2020036