Corresponding author: Evgeny G. Kulikov (

Academic editor: Yury Korovin

The paper considers the applicability of small perturbation theory to assessing the variations of the prompt neutron lifetime caused by variations in the isotope composition of a lead-cooled fast reactor. The generalized small perturbation theory formulas have been developed to calculate derivatives of the prompt neutron lifetime regarded as a bilinear neutron flux and neutron worth ratio. A numerical algorithm has been proposed for the step-by-step application of the small perturbation theory formulas to assess the prompt neutron lifetime variations caused by a major perturbation in the reactor isotope composition, e.g. by the complete change of the material used earlier as the neutron reflector. The advantage of the proposed approach has been shown which consists in that it is basically possible to determine the role of different neutron reactions, isotopes and energy groups in and their contributions to the total prompt neutron lifetime variation caused by major changes in the reactor isotope composition.

Neutronic and optimization calculations of nuclear reactors involve the need for comparing a variety of alternative designs in terms of multiple parameters that are important to ensuring efficient and safe operation of the reactor. These parameters include effective multiplication factor, breeding ratio, temperature and density reactivity coefficients, prompt neutron lifetime, effective fraction of delayed neutrons, etc. Some of these parameters can be presented as bilinear functionals of the neutron flux density of the following type:

where

Some of the other parameters may have the form of bilinear functionals of the neutron flux density and worth as shown below:

where φ^{+}(

This paper deals with investigating the possibility for using small perturbation theory to estimate the prompt neutron lifetime variations as a bilinear functional of the neutron flux density and neutron worth in the event of major perturbations in the reactor isotope composition. An example was considered involving the complete change of the neutron reflector material in the course of which natural lead was changed for the material with a very small neutron absorption capacity. This is radiogenic lead with a high content of the ^{208}Pb isotope.

Some publications (

An attempt was made in

The reactor flux density distribution equation can be written in the following operator form:

where _{eff} is the effective multiplication factor.

For a unidimensional model of a nuclear reactor and multi-group diffusion approximation, equation (1) and its operators

where α is the geometry indicator; _{cdf} is the neutron capture, fission and moderation cross-section; ω_{z}^{2} is the geometrical parameter; χ is the fission neutron spectrum; ν×Σ_{f} is the fission neutron generation macroscopic cross-section; and Σ^{m→k} is the neutron intergroup transition macroscopic cross-section.

According to small perturbation theory (

where ρ_{l}_{,i} is the concentration of the

The equation that describes the energy spatial distribution of the neutron worth is similar to equation (1) and differs from that in using Lagrange conjugate operators, i.e.

An important advantage of formula (2) is that it allows estimating the contribution of different processes (interzone and intergroup transitions of neutrons, and neutron absorption, moderation and multiplication), different isotopes and energy groups to the full value of the derivative (sensitivity) _{eff}_{eff}

There is a generalized small perturbation theory for linear and bilinear functionals that allows variations of functionals to be quantified with comparatively small changes of the isotope composition.

The key ideas of generalized small perturbation theory for bilinear functional (

General formalism suggests that an extended functional (Lagrangian operator) is built, in which equations describing the neutron flux density and value distributions are added to the sought-after functional through Lagrangian multipliers.

Equations can be obtained from the condition of the Lagrangian stationarity for Lagrange multipliers and the small perturbation theory formula. These equations have the following form:

where the functional derivatives in the right-hand members can be calculated as follows:

It can be easily shown that the functional derivatives are orthogonal with respect to the solutions of respective homogeneous equations, that is

According to the Fredholm alternative, it follows from this that solutions exist for heterogeneous equations (4) and (5). This results in the following two systems of homogeneous and heterogeneous equations.

Homogeneous equations:

Heterogeneous equations:

These systems need to be solved successively since the functional derivatives in the right-hand members of heterogeneous equations (6) can be determined only by finding the solution to homogeneous equations (7).

If the systems of homogeneous and heterogeneous equations (6) and (7) are solved, the Lagrangian variance caused, i.e. by the variance of the

It was proved in the functional analysis theory (^{+}(_{J}_{J}^{+}

Therefore, it is expression (8) that is the small perturbation theory formula for the sought-after bilinear functional.

The described procedure for calculating the sensitivity of bilinear functionals was put in the TIME26 code (

The key objective of the calculations was to estimate the variation of this functional with the complete change of the side shield material (natural lead) for lead-208 using small perturbation theory formula (8). The prompt neutron lifetime is specific as a bilinear functional in that the whole of the contribution to its variation is provided only by the terms in expression (8) that describe the neutron leakage, absorption and moderation processes. The small perturbation theory formula (8) can be used to produce the prompt neutron lifetime derivative _{p}_{l, i}, that is, from the concentration of the

In more details, the components in expression (9) can be written as:

By summing up the components in these expressions, one can identify the contributions of individual processes to the prompt neutron lifetime derivative:

contribution of the neutron radial interzone transitions and leakage:

contribution of the axial neutron leakage:

contribution of the neutron absorption:

contribution of the neutron moderation (spectral effect):

If desired, expression (9) can be detailed further and the contributions of individual geometrical zones and energy groups to the prompt neutron lifetime derivative can be determined.

Fig.

Prompt neutron lifetime as a function of the ^{208}Pb fraction in the Pb reflector.

It can be seen from the above curves that the most intensive growth in the prompt neutron lifetime _{p}_{p}_{p}

The explanation is that natural lead as such contains much lead-208 (52.4%). Therefore, the full content of lead-208 in the reflector increases insignificantly at the initial change stage. It is only at the closing stage, when the fraction of lead-208 tends to 100%, that the _{p}

The use of small perturbation theory (9) makes it possible to assess the role played by the neutron interzone transitions, absorption and moderation in the growth of _{p}

A possibility however exists for overcoming this difficulty. The process of substituting natural lead for lead-208 can be broken down into such number of stages within each of which the small perturbation theory formulas predict the variation of _{p}_{p}_{p}

Figs

Contributions of neutron leakage, absorption and moderation to lifetime prolongation (reflector thickness 0.5 m).

Contributions of neutron leakage, absorption and moderation to lifetime prolongation (reflector thickness 1 m).

By summing up the neutron lifetime prolongations for stages of the natural lead gradual substitution for lead-208, one can estimate the contributions of the neutron radial interzone transition, absorption and moderation processes to the complete neutron lifetime prolongation. These data are presented in Table

Contributions of the neutron interzone transition, absorption and moderation to the lifetime prolongation.

Reflector thickness, cm | Δ_{p}(j_{r}), % |
Δ_{p}(Σ_{c}), % |
Δ_{p}(Σ_{d}), % |
---|---|---|---|

50 | 17.9 | 51.6 | 30.5 |

100 | 1.6 | 97.4 | 1.0 |

150 | 0.09 | 101.23 | –1.32 |

200 | –0.003 | 100.607 | –0.604 |

It can be seen from these results that, as the reflector thickness increases, the role of very small neutron absorption by lead-208 increases and becomes rapidly dominant. Interestingly, the cumulative action of the interzone transitions and the spectral effect is insignificant but shortens the prompt neutron lifetime as the reflector thickness is increased.

A stepwise algorithm has been proposed for using small perturbation theory formulas to estimate the sensitivity of the prompt neutron lifetime to major changes in the nuclear reactor isotope composition. The applicability of such approach has been considered for estimating the change in the prompt neutron lifetime caused by the complete substitution of the reflector material in a lead-cooled fast reactor (natural lead was substituted by radiogenic lead with the dominant fraction of the lead-208 isotope). The gradual stepwise substitution of the reflector material has shown that it is possible to use the generalized small perturbation theory formulas at each step and obtain the final result with a good accuracy (with complete substitution of the reflector). The key advantage of the proposed approach is its capability to identify the role and estimate the contribution of all neutron processes, isotopes and energy groups to the complete change of the prompt neutron lifetime.

The work was carried out within the framework of state assignment (project FSWU-2022-0016) with the support of the Ministry of Science and Higher Education of the Russian Federation.

* Russian text published:Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2023, n. 1, pp. 33–43.