CN114912064A - Continuous energy determinism neutron transport calculation method based on function expansion - Google Patents
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Abstract
The invention discloses a neutron transport calculation method based on function expansion continuous energy determinism, which divides an energy region into a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region according to the characteristics of a microscopic cross section of nuclear reaction; each energy interval is divided into a plurality of energy sections; selecting different basis functions in different energy intervals as required, performing function expansion on the neutron angular flux density in different energy intervals, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space; solving the equation in different energy intervals to obtain neutron angular flux density coefficient moment, neutron source strong expansion coefficient moment and neutron angular flux density function of each energy section in the distinguishable resonance energy interval, further realizing the depiction of various nuclear reaction rates of various nuclides, and quantitatively describing the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity, the irradiation environment parameters and the like.
Description
Technical Field
The invention relates to the field of neutron transport process simulation calculation and reactor physical analysis, in particular to a neutron transport calculation method in a continuous energy determinism.
Background
The fission nuclear reactor is a device for carrying out controllable chain type fission reaction, can provide energy, radioactive isotope, neutron beam current, irradiation environment and the like, and is used for industrial production and scientific research. In the core active area, in order to depict the transport process of neutrons in a medium, a neutron transport equation needs to be solved. The mathematical physical equation is a differential-integral equation of neutron angular flux density in a six-dimensional phase space including a three-dimensional space, one-dimensional neutron energy, a two-dimensional flight direction and the like, and belongs to a linear Boltzmann transport equation. The analytical solution of the equation can be obtained only under the condition of very simplification, and the method has important theoretical analysis value and does not have engineering practical significance. Therefore, the physical engineering calculations of the nuclear reactor core are generally performed by a computer using numerical calculation methods. The numerical solution of the neutron transport equation at the present stage is mainly divided into two main methods, one is a determinacy method based on grid division and function expansion, and the average neutron flux density in a finite phase space grid is used for approximating a continuously distributed neutron flux density distribution function in an actual physical problem; the other type is a probability theory method based on random sampling, also called a Monte Carlo method, which converts the neutron flux density into an integral form expected by random process statistics, and obtains an estimated value of the neutron flux density distribution function of the actual physical problem through a large amount of random sampling and statistical estimation.
For the monte carlo method, a reliable result can be given if and only if the number of sample contributions of the target physical quantity is sufficiently large; especially for distributed physical quantities, the computational cost thereof will increase dramatically; in addition, when the finely distributed physical quantity of the large-scale problem is calculated, the method also faces the problem of pseudo convergence or even non-convergence possibly occurring in theory. Therefore, the method is not suitable for massive engineering calculation under the structural geometry. In the determinism method, the calculation efficiency of the structure geometry is high, and a very mature numerical discrete and iterative solution technology has been developed aiming at the joint phase space of the space and the angle of the neutron transport equation, wherein the approximation is less and less. However, in sharp contrast to spatial and angular phase spaces, the conventional multi-cluster approach is still currently used to approximate neutron energy phase spaces. In the multi-group approximation treatment, the whole neutron energy range can be divided into a plurality of (G) intervals, which are called as G energy groups; the probability of all neutrons interacting with the atomic nucleus is considered to be the same in each energy cluster, and the probability is called the average microscopic section of the energy cluster; the average microscopic section of the non-resonance energy group is obtained according to the approximate energy spectrum and the evaluation nuclear database, and then the resonance self-screen calculation is carried out in the resonance energy group to obtain the average microscopic section of the resonance energy group, so that the real neutron transport simulation calculation can be carried out. The method is characterized in that an original equation containing continuous energy is discretized into a solving problem of G energy group equations by a multi-group approximation method, group flux data in each energy group can be obtained by iterative solution of the energy group equations, and the number of the energy groups is determined according to the property and the precision requirement of a researched problem.
However, neutron transport process simulation based on multi-cluster approximation techniques faces a series of problems. On one hand, typical neutron energy spectrums are required to be given in advance in the manufacturing of the multi-group database, and cannot be related to specific problems; on the other hand, the resonance self-screen calculation needs to consider various complex influence effects and also comprises a large number of approximate correction technologies, so that the application range and the correction effect of the technology are very limited, and the design analysis work of the nuclear energy device is severely limited.
In order to thoroughly eliminate the problems caused by multi-group approximation and improve the calculation precision and the calculation efficiency, the invention provides a continuous energy determinism neutron transport calculation method based on function expansion directly from a continuous energy nuclear database, can directly and fundamentally eliminate the multi-group database manufacturing and resonance calculation operation, and provides more accurate data support for the design and analysis of a nuclear energy device.
Disclosure of Invention
In order to overcome the problems in the prior art, the invention aims to provide a function expansion-based neutron transport calculation method in the continuous energy determinism, which can directly and fundamentally eliminate the operations of multi-group database manufacturing and resonance calculation and provide more accurate data support for the design and analysis of a nuclear energy device.
In order to achieve the purpose, the invention adopts the following calculation scheme to implement:
a neutron transport calculation method based on a function expansion continuous energy determinism comprises the following steps:
step 1: the energy region is divided into the following parts according to the microscopic cross section characteristics of nuclear reaction: a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region; each energy interval can be divided into a plurality of energy sections, and the denser the division of the energy sections is, the higher the calculation precision is;
and 2, step: selecting different basis functions R according to requirements in different energy intervals n (u), N being 0,1,2, …, N, and developing a function of the neutron angular flux density over different energy intervals, the neutron angular flux density being developed as a sum of products of unknown coefficients and different order basis functions, as follows:
phi (r, omega, u) -neutron angular flux density (cm) at space r along angle omega with log energy of flight energy u -2 s -1 ) (ii) a Wherein u is lnE, and E is the flight energy of the emergent neutron, and the unit is: MeV;
ψ n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;
selected basis functions R in different energy intervals n (u) accurately characterizing the variation of neutron angular flux density with energy fluctuation within the energy interval; the greater the expansion order N of the basis function is, the more the description precision of different energy regions can be met;
and step 3: corresponding to the fast neutron energy area, the nuclear reaction microscopic section continuously and smoothly changes along with the incident neutron energy, so that the corresponding neutron angular flux density also shows the trend of continuous and smooth change; aiming at the characteristics, selecting a smoothly-changed basis function, only considering a self-scattering source item of an energy area (no higher energy area is scattered to a scattering source item under the energy area), performing basis function expansion on neutron angular flux density of a fast neutron energy area, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to a neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy area function expansion is in a format of a formula (1), and solving the equation to obtain the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy section of a fast neutron energy area; the condition that the energy spectrum needs to be preset in the calculation in a multi-group method is avoided, and the calculation is more accurate;
in the formula:
omega-angle of outgoing neutrons in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic Total Cross-section (cm) of Nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；σ s,k The (u '→ u, omega' → omega) -kth nuclide undergoes a scattering nuclear reaction with a neutron of log energy u 'and a direction of flight omega' to produce a microscopic scattering cross-section (cm) of log energy u and a direction of flight omega neutron 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
and 4, step 4: by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval obtained by calculation in the step 3, the lower scattering source term from the fast neutron energy area to the indistinguishable resonance energy area can be obtained,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
then, self-scattering source items in an indistinguishable resonance energy interval need to be considered, in an indistinguishable resonance energy area, excitation curves of microscopic sections changing along with incident neutron energy have excessively dense formants, the determination value of the specified incident neutron energy microscopic section cannot be determined experimentally, only the probability density of the specified incident neutron energy microscopic section in a certain value range can be given, the probability density is called probability measure, and the probability measure is expressed in the form of a probability table; using a basis function which can accurately measure the distribution of probability while considering the energy distribution, then performing basis function expansion on the neutron angular flux density of an indistinguishable resonance energy region, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy region function expansion is in a formula (2), and solving the equation to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the indistinguishable resonance energy region;
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide is subjected to scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates a microscopic scattering cross section (cm) of neutrons with logarithmic energy u and flight direction omega 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
and 5: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval obtained by calculation in the step 3 can be used for obtaining a lower scattering source item from the fast neutron energy area to the distinguishable resonance energy area,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 3 -distinguishable resonance energy intervals;
the neutron angular flux density coefficient moment of each energy section in the indistinguishable resonance energy interval calculated in the step 4 can be used for obtaining a lower scattering source item from the indistinguishable resonance energy area to the distinguishable resonance energy area,
R 2 n (u) - - - (I) th 2 A base function of the region;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy interval;
in addition, the up scattering source item from the neutron energy region to the distinguishable resonance energy region needs to be heated, an initial value needs to be provided during initial calculation, then up scattering iteration is carried out with the step 6 to obtain an updated up scattering source item,
R 4 n (u) - - - (I) th 4 A base function of the region;
I 4 -thermal neutron energy interval;
I 3 -distinguishable resonance energy interval;
in the distinguishable resonance energy region, the nuclear reaction cross section also has a large number of resonance peaks, the closer to the indistinguishable resonance energy region, the higher the density of the nuclear reaction cross section, the closer to the thermal neutron energy region, the wider the resonance peaks, and the larger the energy range of the nuclear reaction cross section; the nuclear reaction cross section is expressed in a form of a continuous function in a nuclear database; aiming at the characteristics, a basis function capable of accurately representing the abrupt change of the neutron flux density is used, then the neutron angular flux density of a distinguishable resonance energy area is subjected to basis function expansion, and an equation related to the neutron angular flux density in an energy phase space is converted into an equation related to the expansion coefficient moment of the neutron angular flux density in a frequency domain space; the format of a neutron transport equation after the expansion of the basis functions of each energy section of the energy zone is a formula (3), and the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone;
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide is subjected to scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates a microscopic scattering cross section (cm) of neutrons with logarithmic energy u and flight direction omega 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
R 3 n (u) - - - (I) th 3 A base function of the region;
R 4 n (u) - - - (I) th 4 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
step 6: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval obtained by calculation in the step 3 can be used for obtaining a lower scattering source term from the fast neutron energy region to the thermal neutron energy region,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 4 -thermal neutron energy interval;
the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval obtained by calculation in the step 4 can be used for obtaining a lower scattering source term from the indistinguishable resonance energy area to the thermal neutron energy area,
R 2 n (u) - - - (I) th 2 A base function of the region;
I 2 -an indistinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
obtaining a lower scattering source term from the distinguishable resonance energy region to the thermal neutron energy region by using the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy region calculated in the step 5,
R 3 n (u) - - - (I) th 3 A base function of the region;
I 3 -distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
in the thermal neutron energy region, the nuclear reaction microscopic section continuously and smoothly changes along with the incident neutron energy, so that the corresponding neutron flux density also shows the continuous and smooth change trend; aiming at the characteristics, a smooth basis function is selected, then the neutron angular flux density of the thermal energy area is subjected to basis function expansion, and an equation related to the neutron angular flux density in an energy phase space is converted into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space. The format of a neutron transport equation after the expansion of the basis function of each energy section of the energy zone is a formula (4), the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone,
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide undergoes a scattering nuclear reaction with a neutron of log energy u 'and a direction of flight omega' to produce a microscopic scattering cross-section (cm) of log energy u and a direction of flight omega neutron 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
R 3 n (u) - - - (I) th 3 A base function of the region;
R 4 n (u) - - - (I) th 4 A base function of the region;
I 1 -a fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy interval;
I 4 -a thermal neutron energy interval;
and 7: returning to the step 5, forming upper scattering iteration, checking the relative error of the neutron angular flux density coefficient moment of each energy interval obtained by solving before and after each iteration, setting an error limit according to a specific problem, judging calculation convergence if the relative error does not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy zone; if the relative error exceeds the error limit, judging that the convergence is not achieved, and continuing the process from the step 5 to the step 6 until the set calculation iteration number is reached;
and 8: by utilizing the obtained neutron angular flux expansion coefficient moment and the basis functions of the selected areas in different energy intervals, the neutron angular flux density distribution of continuous energy in all the energy intervals can be obtained, so that the depiction of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity and the irradiation environment parameters are quantitatively described.
The invention provides a targeted function expansion technology aiming at the characteristics of nuclear reaction microscopic cross sections in a non-resonance energy area, an indistinguishable resonance energy area and a distinguishable resonance energy area, properly describes the net coupling characteristic of neutrons in an energy space, carries out numerical value dispersion on a neutron transport equation of continuous energy, converts the solution of neutron flux density distribution into the solution of corresponding expansion coefficient moments, and provides the continuous distribution of physical quantities such as neutron flux density, nuclear reaction rate and the like along with neutron energy. And further, the method realizes the depiction of various nuclear reaction rates of various nuclides and provides data support for further quantitatively describing the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity, irradiation environment parameters and the like. The method has the following outstanding advantages: 1) the method eliminates a multi-group database manufacturing link, avoids the problems brought by using a typical neutron energy spectrum in advance, and 2) eliminates a resonance self-screen computing link, and avoids the problems of self-screen and mutual-screen effect correction in space, energy and nuclide composition. 3) The calculation efficiency and the calculation precision can be ensured.
Drawings
FIG. 1 is a schematic diagram of neutron transport calculation based on a function expansion continuum energy determinism.
FIG. 2 is a flow chart of a neutron transport calculation method based on a function expansion continuous energy determinism.
Detailed Description
The invention is described in further detail below with reference to the following figures and specific examples:
as shown in fig. 1, in coordinatesThe curve below the axis shows the neutron angular flux density at a position r and a flight angle omega in any energy interval
As a function of energy, where the solid line represents the true neutron angular flux density that needs to be solved for
The dotted line represents the neutron angular flux density solved for by the actual calculation
The principle of the method of the invention is as follows:
the energy region is divided into the following parts according to the microscopic cross section characteristics of nuclear reaction: a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region, a fast neutron energy region; each energy interval can be divided into a plurality of energy sections. The neutron angular flux density has strong correlation with the nuclear reaction cross section, so that different basis functions R are selected according to requirements in different energy intervals n (u), N is 0,1,2, …, N, and the function expansion is carried out on the neutron angular flux density in different energy intervals, the neutron angular flux density is expanded into the form of the sum of products of unknown coefficients and different order basis functions, and the equation related to the neutron angular flux density in the energy phase space is converted into the equation related to the expansion coefficient moment of the neutron angular flux density in the frequency domain space. Solving the equation in different energy intervals can obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy interval, further realize the depiction of various nuclear reaction rates of various nuclides, and quantitatively describe the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity, the irradiation environment parameters and the like.
As shown in fig. 2, the specific implementation steps of the present invention are as follows:
a neutron transport calculation method based on a function expansion continuous energy determinism comprises the following steps:
step 1: the energy region is divided into the following parts according to the microscopic cross section characteristics of nuclear reaction: a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region; each energy interval can be divided into a plurality of energy sections, and the denser the division of the energy sections is, the higher the calculation precision is;
step 2: selecting different basis functions R according to requirements in different energy intervals n (u), N being 0,1,2, …, N, and developing a function of the neutron angular flux density over different energy intervals, the neutron angular flux density being developed as a sum of products of unknown coefficients and different order basis functions, as follows:
ψ n (r, Ω) -an unknown expansion coefficient moment of neutron angular flux density to be solved;
selected basis functions R in different energy intervals n (u) accurately characterizing the variation of neutron angular flux density with energy fluctuation within the energy interval; the greater the expansion order N of the basis function is, the more the description precision of different energy regions can be met;
and step 3: corresponding to the fast neutron energy area, the nuclear reaction microscopic section continuously and smoothly changes along with the incident neutron energy, so that the corresponding neutron angular flux density also shows the trend of continuous and smooth change; aiming at the characteristics, selecting a smoothly-changed basis function, only considering a self-scattering source item of an energy area (no higher energy area is scattered to a scattering source item under the energy area), performing basis function expansion on neutron angular flux density of a fast neutron energy area, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to a neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy area function expansion is in a format of a formula (1), and solving the equation to obtain the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy section of a fast neutron energy area; the condition that the energy spectrum needs to be preset in the calculation in a multi-group method is avoided, and the calculation is more accurate;
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide is subjected to scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates a microscopic scattering cross section (cm) of neutrons with logarithmic energy u and flight direction omega 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
and 4, step 4: by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval obtained by calculation in the step 3, the lower scattering source term from the fast neutron energy area to the indistinguishable resonance energy area can be obtained,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
then, self-scattering source items in an indistinguishable resonance energy interval need to be considered, in an indistinguishable resonance energy area, excitation curves of microscopic sections changing along with incident neutron energy have excessively dense formants, the determination value of the specified incident neutron energy microscopic section cannot be determined experimentally, only the probability density of the specified incident neutron energy microscopic section in a certain value range can be given, the probability density is called probability measure, and the probability measure is expressed in the form of a probability table; using a basis function which can accurately measure the distribution of probability while considering the energy distribution, then performing basis function expansion on the neutron angular flux density of an indistinguishable resonance energy region, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy region function expansion is in a formula (2), and solving the equation to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the indistinguishable resonance energy region;
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide generates scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates micro dispersion of neutrons with logarithmic energy u and flight direction omegaSection of radiation (cm) 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
and 5: the lower scattering source term from the fast neutron energy region to the distinguishable resonance energy region can be obtained by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 3 -distinguishable resonance energy interval;
the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval calculated in the step 4 can be used for obtaining a lower scattering source item from the indistinguishable resonance energy area to the distinguishable resonance energy area,
R 2 n (u) - - - (I) th 2 A base function of the region;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy interval;
in addition, the up scattering source item from the neutron energy region to the distinguishable resonance energy region needs to be heated, an initial value needs to be provided during initial calculation, then up scattering iteration is carried out with the step 6 to obtain an updated up scattering source item,
R 4 n (u) - - - (I) th 4 A base function of the region;
I 4 -thermal neutron energy interval;
I 3 -distinguishable resonance energy interval;
in the distinguishable resonance energy region, the nuclear reaction cross section also has a large number of resonance peaks, the closer to the indistinguishable resonance energy region, the higher the density of the nuclear reaction cross section, the closer to the thermal neutron energy region, the wider the resonance peaks, and the larger the energy range of the nuclear reaction cross section; the nuclear reaction cross section is expressed in a form of a continuous function in a nuclear database; aiming at the characteristics, a basis function capable of accurately representing the abrupt change of the neutron flux density is used, then the neutron angular flux density of a distinguishable resonance energy area is subjected to basis function expansion, and an equation related to the neutron angular flux density in an energy phase space is converted into an equation related to the expansion coefficient moment of the neutron angular flux density in a frequency domain space; the format of a neutron transport equation after the expansion of the basis functions of each energy section of the energy zone is a formula (3), and the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone;
in the step, on one hand, resonance calculation with large calculation amount is eliminated, the requirement on calculation resources in engineering calculation is greatly reduced, and on the other hand, the energy self-shielding effect, the space self-shielding effect and the mutual shielding effect are eliminated theoretically, so that the calculation is more accurate.
In the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide is subjected to scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates a microscopic scattering cross section (cm) of neutrons with logarithmic energy u and flight direction omega 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
R 3 n (u) - - - (I) th 3 A base function of the region;
R 4 n (u) - - - (I) th 4 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy intervals;
I 4 -a thermal neutron energy interval;
step 6: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval obtained by calculation in the step 3 can be used for obtaining a lower scattering source term from the fast neutron energy region to the thermal neutron energy region,
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -fast neutron energy interval;
I 4 -thermal neutron energy interval;
the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval calculated in the step 4 can be used for obtaining a lower scattering source term from the indistinguishable resonance energy area to the thermal neutron energy area,
R 2 n (u) - - - (I) th 2 A base function of the region;
I 2 -an indistinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
obtaining a lower scattering source term from the distinguishable resonance energy region to the thermal neutron energy region by using the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy region calculated in the step 5,
R 3 n (u) - - - (I) th 3 A base function of the region;
I 3 -distinguishable resonance energy intervals;
I 4 -thermal neutron energy interval;
in the thermal neutron energy region, the nuclear reaction microscopic section continuously and smoothly changes along with the incident neutron energy, so that the corresponding neutron flux density also shows the continuous and smooth change trend; aiming at the characteristics, a smooth basis function is selected, then the neutron angular flux density of the thermal energy area is subjected to basis function expansion, and an equation related to the neutron angular flux density in an energy phase space is converted into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space. The format of a neutron transport equation after the expansion of the basis function of each energy section of the energy zone is a formula (4), the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone,
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k nuclear density (cm) of the kth nuclear species in (r) -space r -3 )；
σ t,k Microscopic total cross-section (cm) of nuclear reaction of (u) -kth nuclide with neutron of logarithmic energy u 2 )；
σ s,k The (u '→ u, omega' → omega) -kth nuclide is subjected to scattering nuclear reaction with neutrons with logarithmic energy u 'and flight direction omega' and generates a microscopic scattering cross section (cm) of neutrons with logarithmic energy u and flight direction omega 2 )；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
R 2 n (u) - - - (I) th 2 A base function of the region;
R 3 n (u) - - - (I) th 3 A base function of the region;
R 4 n (u) - - - (I) th 4 A base function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
and 7: returning to the step 5, forming an upper scattering iteration, checking the relative error of the neutron angular flux density coefficient moment of each energy interval obtained by solving before and after each iteration, setting an error limit according to a specific problem, judging calculation convergence if the relative error does not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy zone; if the relative error exceeds the error limit, judging that the convergence is not achieved, and continuing the process from the step 5 to the step 6 until the set calculation iteration number is reached;
and 8: by utilizing the obtained neutron angular flux expansion coefficient moment and the basis functions of the selected areas in different energy intervals, the neutron angular flux density distribution of continuous energy in all the energy intervals can be obtained, so that the depiction of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity and the irradiation environment parameters are quantitatively described.
Aiming at the characteristics of nuclear reaction microscopic cross sections in a non-resonance energy area, an indistinguishable resonance energy area and a distinguishable resonance energy area, a targeted function expansion technology is provided, the net coupling characteristic of neutrons in an energy space is properly described, a neutron transport equation with continuous energy is subjected to numerical value dispersion, the solution of neutron flux density distribution is converted into the solution of corresponding expansion coefficient moments, and the continuous distribution of physical quantities such as neutron flux density, nuclear reaction rate and the like along with neutron energy is given. And further, the method realizes the depiction of various nuclear reaction rates of various nuclides and provides data support for further quantitatively describing the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity, irradiation environment parameters and the like. The method has the following outstanding advantages: 1) the method eliminates a multi-group database manufacturing link, avoids the problems caused by using a typical neutron energy spectrum in advance, and 2) eliminates a resonance self-screen computing link, and avoids the self-screen and mutual-screen effect correction problems in space, energy and nuclide composition. 3) The calculation efficiency and the calculation precision can be ensured.
On one hand, the method is mainly completed through a computer, so that multiple times of simulation can be simultaneously performed under typical working conditions according to a design scheme, and the time cost is greatly reduced; on the other hand, the experimental analysis of the nuclear energy device is huge in consumption and possibly harms experimental personnel, so that the method can reduce the times of the experimental analysis and reduce the experiment cost which is huge in consumption on the premise of ensuring certain calculation precision. And a design idea can be provided for the design of related nuclear energy devices, and quantitative data can be provided for further improving the performance of the nuclear energy devices.
Claims (1)
1. A neutron transport calculation method based on a function expansion continuous energy determinism is characterized in that: the method comprises the following steps:
step 1: the energy region is divided into the following parts according to the microscopic cross section characteristics of nuclear reaction: a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region; each energy interval can be divided into a plurality of energy sections;
step 2: selecting different basis functions R according to requirements in different energy intervals n (u), N being 0,1,2, …, N, and developing a function of the neutron angular flux density over different energy intervals, the neutron angular flux density being developed as a sum of products of unknown coefficients and different order basis functions, as follows:
phi (r, omega, u) -the neutron angular flux density in space r along an angle omega with logarithmic energy of flight energy u in units of: cm of -2 s -1 (ii) a Wherein u is lnE, and E is the flight energy of the emergent neutron, and the unit is: MeV;
ψ n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;
selected basis functions R in different energy intervals n (u) accurately depicting the variation of neutron angular flux density with energy fluctuation within the energy interval; the greater the expansion order N of the basis function is, the more the description precision of different energy regions can be met;
and step 3: selecting a smoothly-changed basis function, only considering a self-scattering source item of an instron, performing basis function expansion on neutron angular flux density of a fast neutron energy area, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to a neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy area function expansion is in a format of a formula (1), and solving the equation to obtain the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy section of a fast neutron energy interval;
in the formula:
omega-angle of neutron extraction in space;
Ω' - -angle of incident neutrons in space;
r- -spatial position;
u-log energy of the emitted neutron energy;
u' -log energy of incident neutron energy;
N k (r) -nuclear density of the kth nuclear species in space r in units of: cm of -3 ；
σ t,k Microscopic total cross-section of nuclear reaction of (u) -kth nuclide with a neutron of logarithmic energy u in units of: cm of 2 ；
σ s,k (u '→ u, Ω' → Ω) - -kth nuclide scatters nuclei with neutrons having a logarithmic energy u 'and a flight direction Ω' and produces microscopic scattering cross sections with neutrons having a logarithmic energy u and a flight direction Ω, in units of: cm 2 ；
q n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;
R 1 n (u) - - - (I) th 1 A base function of the region;
I 1 -a fast neutron energy interval;
and 4, step 4: obtaining the lower scattering source term from the fast neutron energy region to the indistinguishable resonance energy region by using the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,
R 1 n (u) - - - (I) th 1 Basis functions of regions；
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
then, self-scattering source items in an indistinguishable resonance energy interval need to be considered, in an indistinguishable resonance energy area, excitation curves of microscopic sections changing along with incident neutron energy have excessively dense formants, the determination value of the specified incident neutron energy microscopic section cannot be determined experimentally, only the probability density of the specified incident neutron energy microscopic section in a certain value range can be given, the probability density is called probability measure, and the probability measure is expressed in the form of a probability table; using a basis function which can accurately measure the distribution of probability while considering the energy distribution, then performing basis function expansion on the neutron angular flux density of an indistinguishable resonance energy region, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy region function expansion is in a formula (2), and solving the equation to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the indistinguishable resonance energy region;
in the formula:
R 2 n (u) - - - (I) th 2 A base function of the region;
I 2 -an indistinguishable resonance energy interval;
and 5: obtaining a lower scattering source term from the fast neutron energy region to the distinguishable resonance energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,
I 3 -distinguishable resonance energy interval;
the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval obtained by calculation in the step 4 is utilized to obtain a lower scattering source item from the indistinguishable resonance energy area to the distinguishable resonance energy area,
in addition, the neutron energy region is heated to the upper scattering source item of the distinguishable resonance energy region, the upper scattering source item needs to provide an initial value during initial calculation, then the upper scattering iteration is carried out with the step 6 to obtain an updated upper scattering source item,
R 4 n (u) - - - (I) th 4 A base function of the region;
I 4 -a thermal neutron energy interval;
using a basis function capable of accurately representing the abrupt change of the neutron flux density, then performing basis function expansion on the neutron angular flux density of a distinguishable resonance energy region, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the expansion coefficient moment of the neutron angular flux density in a frequency domain space; the format of a neutron transport equation after the expansion of the basis functions of each energy section of the energy zone is a formula (3), and the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone;
in the formula:
R 3 n (u) - - - (I) th 3 A base function of the region;
step 6: obtaining a lower scattering source term from the fast neutron energy region to the thermal neutron energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,
obtaining the lower scattering source term from the indistinguishable resonance energy region to the thermal neutron energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy region obtained by calculation in the step 4,
obtaining a lower scattering source term from the distinguishable resonance energy region to the thermal neutron energy region by using the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy region calculated in the step 5,
selecting a smooth basis function, then performing basis function expansion on the neutron angular flux density of the thermal energy region, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space. The format of the neutron transport equation after the basis functions of each energy section of the energy zone are expanded is formula (4), the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the identifiable resonance energy zone,
and 7: returning to the step 5, forming an upper scattering iteration, checking the relative error of the neutron angular flux density coefficient moment of each energy interval obtained by solving before and after each iteration, setting an error limit according to a specific problem, judging calculation convergence if the relative error does not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy zone; if the relative error exceeds the error limit, judging that the convergence is not achieved, and continuing the process from the step 5 to the step 6 until the set calculation iteration number is reached;
and 8: the neutron angular flux expansion coefficient moment and the basis function of the selected area in different energy intervals are utilized to obtain the neutron angular flux density distribution of continuous energy in all the energy intervals, so that the depiction of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity and the irradiation environment parameters are quantitatively described.
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