Corresponding author: Ivan V. Tormyshev (itormyshev@ippe.ru)

Academic editor: Georgy Tikhomirov

The article discusses the neutron flux stability in the core of a high-power sodium-cooled fast reactor (of the BN-type) without feedbacks. The importance of this problem for high-power BN-type reactors is associated with the specific features of the layout of their cores, including a large diameter and height/diameter ratio about 5. The technique used to substantiate the stability of neutron fields is based on the analysis of the spectrum of the matrix of the system of spatial kinetics equations describing the core of a high-power BN-type reactor without feedbacks. A computational model of the spatial kinetics of a high-power BN-type reactor has been developed in the modal approximation based on the representation of an unsteady flux as a sum of orthogonal functions multiplied by time-dependent amplitudes. The eigenfunctions of the conditionally critical problem are used in the diffusion approximation, which in the discrete case form a complete system. The spectrum of the matrix of the system of ordinary differential equations describing the spatial kinetics of the reactor has been calculated. It is shown that the neutron flux in the core of a high-power BN-type reactor without feedbacks is stable. Test calculations have illustrated the damping of perturbations of the power distribution for a reactor in a critical state.

Numerous safety studies of sodium-cooled fast reactors have shown that, in order to exclude the possibility of destruction of the core in beyond design basis accidents of the ULOF type, the core height should be no more than 85–90 cm, and above it there should be a sodium plenum (

The concept of ‘stability’ as applied to nuclear reactors is formulated similarly to the concept of stability of any dynamic systems, i.e., the reactor state is called ‘stable’ if the deviations of the reactor parameters (including power) from the stationary values that have arisen after the introduction of a disturbance into the reactor do not increase indefinitely with time (limited response to limited impact). Otherwise, if the disturbance introduced into the reactor increases with time indefinitely, the reactor state is called ‘unstable’ (

Mathematical criteria for the stability of dynamical systems are formulated in the theory of differential equations.

The solution φ(

with the initial conditions

If the solution φ (

then the solution of the system is said to have the property of asymptotic stability (

One of the methods used to assess the stability of power distribution in nuclear reactors is to analyze the stability of a system of differential equations describing the kinetics of a reactor in the approximation of expansion in some system of functions (as a rule, in terms of eigenfunctions of a conditionally critical diffusion problem) (_{i}

In this work, the stability of the power distribution of a high-power BN-type sodium-cooled fast reactor without feedbacks is considered by analyzing the eigenvalues of a system of differential equations describing the reactor kinetics when the neutron flux is expanded in terms of the eigenfunctions of the conditionally critical problem. The nuclear reactor dynamics model without feedbacks is applicable to the analysis of the processes occurring in the reactor core when the reactor operates in the starting power range.

One of the ways to solve the problems of spatial kinetics is to expand the non-stationary neutron flux into a series in terms of the system of orthogonal functions:

As shown in (

Let us consider the non-stationary diffusion equation (coordinates

where

Δ_{j}_{p}_{j}_{j}_{d}_{j}

Let us substitute expansion (1) into system (2), choosing as functions Φ_{i}

_{i}_{i}_{i}

Taking into account the equalities _{i}_{i}_{i}_{i}_{p} F_{i}_{d} F_{i}_{d}

.

We multiply the first equation of the resulting system by the eigenfunctions Φ^{+}_{k}

^{+}Φ^{+}_{k}^{+}Φ^{+}_{i}_{k}

and the equations for delayed neutrons by χ_{d}_{j}^{+}_{k}

Taking into account the orthogonality of the eigenfunctions of the direct and adjoint problems

(here γ_{0} is the fission neutrons importance, for _{k}

By entering parameters

ρ_{k}_{k}

Λ_{k}_{i}^{+}_{k}_{i}^{+}_{k}_{k}

β_{eff}_{k}_{i}_{j}^{+}_{k}_{d}_{j}_{j}F_{i}^{+}_{k}_{k}

β_{eff}_{k}_{i}^{+}_{k}_{d}_{i}^{+}_{k}_{k}

_{k}_{j}^{+}_{k}_{d}_{j}C_{j}^{+}_{k}_{k}

_{k}^{+}_{k}^{+}_{k}_{k}

δ_{k}_{i}^{+}_{k}_{i}^{+}_{k}_{k}

δ_{k}_{i}^{+}_{k}_{i}^{+}_{k}_{k}

δ_{k}_{i,j}^{+}_{k}_{d}_{j}_{i}^{+}_{k}_{k}

which are a generalization of the parameters of point kinetics, we bring the system to the form

Here _{k}_{i}_{k}_{eff}_{k}_{i}_{j}_{eff}_{k}_{i}_{j}_{j}_{j}_{k}_{j}_{k}_{j}_{k}_{k}_{i}_{k}_{i}_{j}_{k}_{i,j}

The number of equations in system (9) is determined by the number of eigenfunctions used to represent the spatial dependence of the neutron flux _{d}_{d}

In order to check the correctness of the equations obtained, a calculation was performed using the modal kinetics model of the transient process in a high-power BN-type reactor.

To calculate the coefficients included in system (9), and to solve this system, computer programs were developed.

The higher harmonics of the conditionally critical problem, necessary for calculating the coefficients of system (9), were calculated in the diffusion approximation using the modified neutronic calculation module of the REACTOR software package (

The parameters of system (9) were calculated by summing over the volume of the calculated region the values of the direct and adjoint flux harmonics, multiplied by the corresponding coefficients, i.e., the inverse neutron velocity, fission neutron source, delayed neutron source, external neutron source, deviations of the neutron transfer operator and the fission neutron source and delayed neutrons from a stationary state.

To solve system of equations (9), the software package for solving systems of ordinary differential equations ODEPACK (

The transient process consisted in the rise of the reactor power after a change in state, leading to the introduction of positive reactivity. In the initial state, the reactor was critical (matrices δ_{j}_{j}_{j}

Conditionally critical calculation of the disturbed state of the reactor showed that the introduced disturbance of the composition corresponded to an increase in reactivity Δρ_{0} = 4,25∙10^{–5}Δ

The transient process was calculated in two models: in the standard model of point kinetics with the assignment of the introduced reactivity according to the law

and in modal kinetics model (9) using 10 eigenfunctions in expansion (1).

The reactor power calculated in the modal approximation (Fig.

Change in the reactor power in the test calculation.

The temporal behavior of all the higher harmonics of the neutron flux during the transient process is characterized by the following regularity: at the initial stage, there is a significant (by several orders of magnitude) increase in the ratio of their amplitudes to the amplitude of the fundamental harmonic |_{i}_{0}(_{i}_{0}(^{–4}) for the fifth harmonic, which is due to the fact that it has a radial character. For the first harmonic, which has an azimuthal character, the maximum value of the ratio |_{i}_{0}(^{–7}.

Relative amplitudes (|_{i}_{0}(

The stability analysis was carried out for the system of equations of the spatial kinetics of an undisturbed reactor without feedbacks.

This system can be rearranged into the form

According to the theory of ordinary differential equations, the stability of the solution to a system of linear differential equations with constant coefficients is determined by the spectrum of the system matrix.

In the case under consideration, the system matrix takes the form

The absence in the spectrum of the matrix of eigenvalues with a positive real part is a criterion for the stability of the system of ordinary differential equations. If all the eigenvalues have negative real parts, the system has asymptotic stability, if among the eigenvalues there is a simple eigenvalue with zero real part, the system is Lyapunov stable (

To analyze the stability of the power distribution in the high-power BN-type reactor, parameters (9) and matrix elements (12) were calculated.

The stability analysis was carried out for two options, i.e., stationary operation at a power level close to the minimum controllable power, when the power of the subcritical reactor is maintained by its own neutron source associated with the decay of transuranic elements in the core, and for stationary operation in a critical state.

The thermal power of the reactor with the source was taken to be 1 kW. According to the estimates made, for the considered version of a high-power BN reactor, this power is achieved at a reactivity of ρ_{0} = –5,51 × 10^{–4}Δ

When we use five terms in expansion (1) and six groups of delayed neutrons, the dimension of system (10) is 35. Thus, matrix (12) has 35 eigenvalues. To solve the complete eigenvalue problem for this matrix, the LAPACK software package (^{–3}s^{–1}. Thus, the system of spatial kinetics equations for a high-power BN-type reactor with a source is asymptotically stable (

To test the effect on the eigenvalues of the system of kinetic equations of changes in the number of harmonics used to describe the spatial dependence of the neutron flux in expansion (1), the matrix spectrum of the system (11) was calculated using one term in the expansion, which corresponds to the approximation of point kinetics. Table

Spectra of matrices of the system of kinetic equations of a high-power BN-type reactor with a source

Point kinetics, s^{–1} |
Spatial kinetics, s^{–1} |
---|---|

–9.28278E–3 | –9.28278E–3 |

–1.68761E–2 | –1.33580E–2 |

–8.05185E–2 | –1.33598E–2 |

–2.01789E–1 | –1.33948E–2 |

–7.43827E–1 | –1.33950E–2 |

–2.78100E+0 | –1.68761E–2 |

–9.83543E+3 | –2.99014E–2 |

–2.99298E–2 | |

–3.05003E–2 | |

–3.05010E–2 | |

–8.05185E–2 |

For the case of point kinetics, the number of eigenvalues is by one greater than the number of groups of delayed neutrons. In the case of expansion in harmonics, each eigenvalue of the system of point kinetics equations corresponds to five (in the considered case) eigenvalues, since the system includes the concentrations of precursor nuclei and the neutron flux expanded in the first five harmonics of the neutron flux. The eigenvalue minimum in absolute value of these five corresponds to the expansion of the flux and the concentration of precursor nuclei in terms of the fundamental harmonic and coincides with the corresponding eigenvalue of the point kinetics system. Thus, the first eigenvalue of the system written for the expansion with five terms coincides with the first eigenvalue of the system of point kinetics equations, the sixth one coincides with the second eigenvalue of the point kinetics system, the 11^{th} eigenvalue coincides with the third, etc.

As can be seen from the table, refining the description of the reactor kinetics does not lead to an increase in the maximum eigenvalue of the system of reactor kinetics equations.

For a reactor operating in a critical state at ρ_{0} = 0, the spectrum of the matrix of point kinetics system and seven maximum eigenvalues of the matrix of spatial kinetics system are given in Table

Spectra of matrices of the system of kinetic equations of a high-power BN-type reactor in a critical state

Point kinetics, s^{–1} |
Spatial kinetics, s^{–1} |
---|---|

0.0 | 0.0 |

–1.5019E–2 | –1.3356E–2 |

–7.0303E–2 | –1.3358E–2 |

–1.8911E–1 | –1.3395E–2 |

–7.2721E–1 | –1.3395E–2 |

–2.7576E+0 | –1.5019E–2 |

–8.5260E+3 | –2.9874E–2 |

The maximum eigenvalue of kinetics system of a critical high-power BN-type reactor without feedbacks is zero; thus, its power distribution is Lyapunov stable (

Calculations of the decay of disturbances of the first and fourth higher harmonics of the neutron flux were performed. After a disturbance equal to 10 W is introduced, the amplitude of the harmonic decreases to a value close to zero in a time of the order of 10^{–4} s (Fig.

Decay of disturbances of the first and fourth higher harmonics in a high-power BN-type reactor.

As shown by the calculation of the spectra of the matrices of the system of differential spatial kinetics equations for the subcritical and critical states of the high-power BN-type reactor without feedbacks, the stability criterion for the states considered is fulfilled, the neutron flux in the subcritical core of the BN reactor is asymptotically stable, the neutron flux in the critical core of the BN reactor is Lyapunov stable.

We plan to continue our work by adding feedbacks to the spatial kinetics model and substantiating the stability of the neutron flux in the high-power BN core with feedbacks.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2021, n. 2, pp. 39–49.