CN114912064B - Continuous energy certainty theory neutron transport calculation method based on function expansion - Google Patents
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Abstract
The invention discloses a continuous energy deterministic neutron transport calculation method based on function expansion, which divides an energy zone into a thermal neutron energy zone, a distinguishable resonance energy zone, an indistinguishable resonance energy zone and a fast neutron energy zone according to the microcosmic section characteristics of nuclear reaction; wherein each energy interval is divided into a plurality of energy sections; selecting different basis functions according to requirements in different energy intervals, performing function expansion on neutron angular flux densities in different energy intervals, and converting an equation about neutron angular flux densities in an energy phase space into an equation about 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 and neutron source strong expansion coefficient moment and neutron angular flux density functions of each energy section of the distinguishable resonance energy interval, further realizing the depiction of various nuclear reaction rates of various nuclides, and quantitatively describing the release of the fission energy, the consumption and production of the nuclides, the neutron beam intensity, 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 continuous energy definite neutron transport calculation method.
Background
The fission reactor is a device for carrying out controllable chain fission reaction, and can provide energy, radioactive isotopes, neutron beam, irradiation environment and the like for industrial production and scientific research. In order to describe the neutron transport process in the medium in the reactor core active region, a neutron transport equation needs to be solved. The mathematical physical equation is a differential-integral equation of neutron angular flux density in six-dimensional phase space such as three-dimensional space, one-dimensional neutron energy, 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 simplicity, has important theoretical analysis value, and has no engineering practical significance. Therefore, the physical engineering calculation of the nuclear reactor core is generally completed by a computer by using a numerical calculation method. The numerical solution of the neutron transport equation at the present stage is mainly divided into two main methods, one is a determination method based on grid division and function expansion, and the average neutron flux density in a limited phase space grid is used for approaching a neutron flux density distribution function which is continuously distributed in an actual physical problem; the other is a probability theory method based on random sampling, also called a Monte Carlo method, which converts neutron flux density into an integral form expected by random process statistics, and an estimated value of a neutron flux density distribution function of an actual physical problem is obtained through a large number 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 will increase dramatically; in addition, when the fine distribution physical quantity of the large-scale problem is calculated, the method also faces the problem of false convergence or even non-convergence which possibly occurs in theory. Therefore, under the structural geometry, the method is not suitable for mass engineering calculation. In the deterministic approach, the computational efficiency of structural geometry is high, and very sophisticated numerical discrete and iterative solution techniques have been developed for the joint phase space of space and angle of neutron transport equations, with less and less approximations. However, in sharp contrast to space and angular phase space, conventional multi-cluster approaches are still currently being used to approximate neutron energy phase space. In a multi-cluster approximation process, the entire neutron energy range may be divided into several (G) intervals, referred to as G energy clusters; the probability of all neutrons interacting with the nuclei is considered to be the same within each energy cluster, referred to as the average microscopic cross section of that energy cluster; the real neutron transport simulation calculation can be performed by acquiring the average microscopic section of the non-resonance energy group according to the approximate energy spectrum and the evaluation core database, and then performing resonance self-screen calculation in the resonance energy group to acquire the average microscopic section of the resonance energy group. The method is characterized in that equations originally containing continuous energy are discretized into solving problems of G energy group equations through a multi-group approximation method, the iteration solving of the energy group equations can be used for obtaining group flux data in each energy group, and the number of the energy groups is determined according to the property and the precision requirements of the studied problems.
However, neutron transport process simulation based on the multi-group approximation technique faces a series of problems. On the one hand, the preparation of the multi-group database requires the presetting of typical neutron energy spectrum, and cannot be related to specific problems; on the other hand, the resonance self-shielding calculation needs to consider complex and various influence effects, and a large number of approximate correction technologies are included, 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 directly starts from a continuous energy nuclear database, provides a continuous energy definite theory neutron transport calculation method based on function expansion, can directly and fundamentally remove multi-group database manufacturing and resonance calculation operation, and provides more accurate data support for the design 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 continuous energy definite theory neutron transport calculation method based on function expansion, which can directly and fundamentally eliminate multi-group database manufacturing and resonance calculation operations and provides more accurate data support for design analysis of a nuclear energy device.
In order to achieve the above purpose, the invention adopts the following calculation scheme to implement:
a continuous energy definite neutron transport calculation method based on function expansion comprises the following steps:
step 1: the energy region is divided into the following steps according to the microcosmic 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 energy sections are, the higher the calculation accuracy is;
step 2: different basis functions R are selected according to the need in different energy intervals n (u), n=0, 1,2, …, N, and functionally expanding the neutron angular flux density over different energy intervals to a form of the sum of products of unknown coefficients and different order basis functions, as follows:
phi (r, omega, u) -neutron angular flux density (cm) flying at angle omega at space r and logarithmic energy of flight energy u -2 s -1 ) The method comprises the steps of carrying out a first treatment on the surface of the Where u= lnE, E is the outgoing neutron flight energy in units of: meV;
ψ n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;
base function R selected in different energy intervals n (u) can accurately describe the fluctuation of neutron angular flux density along with energy in the energy interval; the larger the expansion order N of the basis function is, the more the depiction precision of different energy areas can be met;
step 3: corresponding to a fast neutron energy region, the microscopic cross section of nuclear reaction continuously and smoothly changes along with the energy of the incident neutrons, so that the corresponding neutron angular flux density also shows a trend of continuous and smooth change; according to the characteristics, a smooth-change basis function is selected, only a self-scattering source item of an instinct area (a scattering source item under an energy area without higher energy area is considered), an equation about neutron angular flux density in an energy phase space is converted into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space by expanding the basis function of neutron angular flux density of the fast neutron energy area, a neutron transport equation format after expansion of the energy area function is the formula (1), and the equation is solved to obtain neutron angular flux density coefficient moment and neutron angular flux density functions of each energy section of the fast neutron energy area; the condition that the energy spectrum needs to be preset in the multi-group method is avoided, and the calculation is more accurate;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
step 4: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source item from the fast neutron energy region to the indistinguishable resonance energy region,
R 1 n (u) -I 1 A basis function of the region;
I 1 fast neutron energy interval；
I 2 -an indistinguishable resonance energy interval;
then, considering a self-scattering source item in an indistinguishable resonance energy interval, wherein an excitation curve of a microscopic section along with the change of the incident neutron energy in the indistinguishable resonance energy interval is provided with excessively dense resonance peaks, a determined value of the microscopic section of the appointed incident neutron energy cannot be determined experimentally, and only the probability density of the determined value in a certain value range can be given, which is called probability measure, and the probability measure is expressed in a probability table form; the method comprises the steps of using a basis function capable of considering energy distribution and simultaneously accurately measuring probability distribution, then performing basis function expansion on neutron angular flux density of an indistinguishable resonance energy region, converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation format after the energy region function expansion is formula (2), and solving the equation to obtain neutron angular flux density coefficient moment and neutron source strong expansion coefficient moment of each energy section of the indistinguishable resonance energy region and neutron angular flux density function;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth species react with neutrons with logarithmic energy u ', flight direction Ω' to produce logarithmic numbersMicrocosmic scattering cross section (cm) of neutrons with energy u and flight direction omega 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
step 5: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source item from the fast neutron energy region to the distinguishable resonance energy region,
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
I 3 -a 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 is utilized to obtain the lower scattering source term from the indistinguishable resonance energy zone to the distinguishable resonance energy zone,
R 2 n (u) -I 2 A basis function of the region;
I 2 -an indistinguishable resonance energy interval;
I 3 -a distinguishable resonance energy interval;
in addition, the thermal neutron energy region to the distinguishable resonance energy region is needed to be provided with an initial value in the initial calculation, then the up-scattering iteration is carried out with the step 6 to obtain an updated up-scattering source item,
R 4 n (u) -I 4 A basis function of the region;
I 4 -thermal neutron energy interval;
I 3 -a distinguishable resonance energy interval;
in the distinguishable resonance energy region, the nuclear reaction section also has a large number of formants, the closer to the indistinguishable resonance energy region, the higher the density of the nuclear reaction section is, the closer to the thermal neutron energy region, the wider the formants are, and the larger the energy range of influence is; the nuclear reaction cross section is represented in the nuclear database in the form of a continuous function; aiming at the characteristics, a basis function capable of accurately representing abrupt changes of neutron flux density is used, then the basis function expansion is carried out on neutron angular flux density of a distinguishable resonance energy region, and an equation about neutron angular flux density in an energy phase space is converted into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space; the neutron transport equation format after the base function of each energy segment of the energy region is expanded is formula (3), and the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy segment of the distinguishable resonance energy region are obtained by solving the equation;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
R 3 n (u) -I 3 A basis function of the region;
R 4 n (u) -I 4 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -a 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 calculated in the step 3 is utilized to obtain the lower scattering source items from the fast neutron energy region to the thermal neutron energy region,
R 1 n (u) -I 1 A basis 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 is utilized to obtain the lower scattering source term from the indistinguishable resonance energy zone to the thermal neutron energy zone,
R 2 n (u) -I 2 A basis function of the region;
I 2 -an indistinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy interval calculated in the step 5 is utilized to obtain a lower scattering source item from the distinguishable resonance energy zone to the thermal neutron energy zone,
R 3 n (u) -I 3 A basis function of the region;
I 3 -a distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
in a thermal neutron energy region, the microscopic cross section of nuclear reaction continuously and smoothly changes along with the energy of the incident neutrons, so that the corresponding neutron flux density also shows a trend of continuous and smooth change; according to the characteristics, a smooth basis function is selected, then the basis function expansion is carried out on the neutron angular flux density of the thermal energy region, and an equation about the neutron angular flux density in the energy phase space is converted into an equation about the neutron angular flux density expansion coefficient moment in the frequency domain space. The neutron transport equation format after the expansion of the basis functions of all the energy sections of the energy region is the formula (4), the neutron angular flux density coefficient moment and the neutron source strong expansion coefficient moment and the neutron angular flux density function of all the energy sections of the distinguishable resonance energy region are obtained by solving the equation,
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
R 3 n (u) -I 3 A basis function of the region;
R 4 n (u) -I 4 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -a distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
step 7: returning to the step 5, forming up-scattering iteration, then checking the relative errors 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 that calculation converges if the relative errors do not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy interval; if the relative error exceeds the error limit, judging that the relative error is not converged, and continuing the processes from the step 5 to the step 6 until the set calculation iteration times are reached;
step 8: the obtained neutron angular flux expansion coefficient moment and the basis functions of selected areas in different energy intervals are utilized to obtain neutron angular flux density distribution of continuous energy in all the energy intervals, so that the description of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of nuclides, neutron beam intensity and irradiation environment parameters are quantitatively described.
Aiming at the characteristics of a nuclear reaction microscopic section in a non-resonance energy region, an indistinguishable resonance energy region and a distinguishable resonance energy region, the invention provides a targeted function expansion technology, properly describes the net coupling characteristic of neutrons in an energy space, performs numerical dispersion on a neutron transport equation of continuous energy, converts the solution of neutron flux density distribution into the solution of corresponding expansion coefficient moment, and gives out the continuous distribution of physical quantities such as neutron flux density, nuclear reaction rate and the like along with the neutron energy. And further, the description of various nuclear reaction rates of various nuclides is realized, and data support is provided for further quantitatively describing the release of the fission energy, the consumption and production of the nuclides, the neutron beam intensity, the irradiation environment parameters and the like. The method has the outstanding advantages that: 1) Eliminating the multi-group database making link, avoiding the problems caused by the prior use of typical neutron energy spectrum, 2) eliminating the resonance self-shielding calculation link, and avoiding the self-shielding and mutual shielding effect correction problems in space, energy and nuclide constitution. 3) The calculation efficiency and the calculation precision can be ensured.
Drawings
FIG. 1 is a schematic diagram of a continuous energy deterministic neutron transport calculation based on a functional expansion.
FIG. 2 is a flow chart of a continuous energy deterministic neutron transport calculation method based on function expansion.
Detailed Description
The invention is described in further detail below with reference to the attached drawings and to specific examples:
as shown in FIG. 1, the curve below the coordinate axis shows the neutron angular flux density at a flight angle Ω at a location r within any energy intervalWith energy, wherein the solid line represents the true neutron angular flux density that needs to be solvedThe dashed line represents the neutron angular flux density of the actual calculated solution>
The method principle of the invention is as follows:
the energy region is divided into the following steps according to the microcosmic 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; wherein each energy interval can be divided into a plurality of energy segments. The neutron angular flux density has strong correlation with the nuclear reaction section, so different basis functions R are selected according to the need in different energy intervals n (u), n=0, 1,2, …, N, and functionally expanding neutron angular flux densities in different energy intervals, expanding neutron angular flux densities into the form of the sum of products of unknown coefficients and different order basis functions, and converting equations in energy phase space about neutron angular flux densities into equations in frequency domain space about neutron angular flux density expansion coefficient moments. Solving the equation in different energy intervals can obtain neutron angular flux density coefficient moment and neutron source strong expansion coefficient moment and neutron angular flux density functions of each energy section in the distinguishable resonance energy interval, and further realize the depiction of various nuclear reaction rates of various nuclides, and quantitatively describe the release of the fission energy, the consumption and production of nuclides, the neutron beam intensity, irradiation environment parameters and the like.
As shown in fig. 2, the specific implementation steps of the present invention are as follows:
a continuous energy definite neutron transport calculation method based on function expansion comprises the following steps:
step 1: the energy region is divided into the following steps according to the microcosmic 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 energy sections are, the higher the calculation accuracy is;
step 2: different basis functions R are selected according to the need in different energy intervals n (u), n=0, 1,2, …, N, and functionally expanding the neutron angular flux density over different energy intervals to a form of the sum of products of unknown coefficients and different order basis functions, as follows:
-neutron angular flux density (cm) of logarithmic energy u flying at angle Ω at space r -2 s -1 ) The method comprises the steps of carrying out a first treatment on the surface of the Where u= lnE, E is the outgoing neutron flight energy in units of: meV;
ψ n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;
base function R selected in different energy intervals n (u) can accurately describe the fluctuation of neutron angular flux density along with energy in the energy interval; the larger the expansion order N of the basis function is, the more the depiction precision of different energy areas can be met;
step 3: corresponding to a fast neutron energy region, the microscopic cross section of nuclear reaction continuously and smoothly changes along with the energy of the incident neutrons, so that the corresponding neutron angular flux density also shows a trend of continuous and smooth change; according to the characteristics, a smooth-change basis function is selected, only a self-scattering source item of an instinct area (a scattering source item under an energy area without higher energy area is considered), an equation about neutron angular flux density in an energy phase space is converted into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space by expanding the basis function of neutron angular flux density of the fast neutron energy area, a neutron transport equation format after expansion of the energy area function is the formula (1), and the equation is solved to obtain neutron angular flux density coefficient moment and neutron angular flux density functions of each energy section of the fast neutron energy area; the condition that the energy spectrum needs to be preset in the multi-group method is avoided, and the calculation is more accurate;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
step 4: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source item from the fast neutron energy region to the indistinguishable resonance energy region,
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
then, considering a self-scattering source item in an indistinguishable resonance energy interval, wherein an excitation curve of a microscopic section along with the change of the incident neutron energy in the indistinguishable resonance energy interval is provided with excessively dense resonance peaks, a determined value of the microscopic section of the appointed incident neutron energy cannot be determined experimentally, and only the probability density of the determined value in a certain value range can be given, which is called probability measure, and the probability measure is expressed in a probability table form; the method comprises the steps of using a basis function capable of considering energy distribution and simultaneously accurately measuring probability distribution, then performing basis function expansion on neutron angular flux density of an indistinguishable resonance energy region, converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation format after the energy region function expansion is formula (2), and solving the equation to obtain neutron angular flux density coefficient moment and neutron source strong expansion coefficient moment of each energy section of the indistinguishable resonance energy region and neutron angular flux density function;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
step 5: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source item from the fast neutron energy region to the distinguishable resonance energy region,
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
I 3 -a 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 is utilized to obtain the lower scattering source term from the indistinguishable resonance energy zone to the distinguishable resonance energy zone,
R 2 n (u) -I 2 A basis function of the region;
I 2 -an indistinguishable resonance energy interval;
I 3 -a distinguishable resonance energy interval;
in addition, the thermal neutron energy region to the distinguishable resonance energy region is needed to be provided with an initial value in the initial calculation, then the up-scattering iteration is carried out with the step 6 to obtain an updated up-scattering source item,
R 4 n (u) -I 4 A basis function of the region;
I 4 -thermal neutron energy interval;
I 3 -a distinguishable resonance energy interval;
in the distinguishable resonance energy region, the nuclear reaction section also has a large number of formants, the closer to the indistinguishable resonance energy region, the higher the density of the nuclear reaction section is, the closer to the thermal neutron energy region, the wider the formants are, and the larger the energy range of influence is; the nuclear reaction cross section is represented in the nuclear database in the form of a continuous function; aiming at the characteristics, a basis function capable of accurately representing abrupt changes of neutron flux density is used, then the basis function expansion is carried out on neutron angular flux density of a distinguishable resonance energy region, and an equation about neutron angular flux density in an energy phase space is converted into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space; the neutron transport equation format after the base function of each energy segment of the energy region is expanded is formula (3), and the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy segment of the distinguishable resonance energy region are obtained by solving the equation;
in the step, on one hand, resonance calculation with larger 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.
Wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density of the kth nuclear species at (r) -space r(cm -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
R 3 n (u) -I 3 A basis function of the region;
R 4 n (u) -I 4 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -a 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 calculated in the step 3 is utilized to obtain the lower scattering source items from the fast neutron energy region to the thermal neutron energy region,
R 1 n (u) -I 1 A basis 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 is utilized to obtain the lower scattering source term from the indistinguishable resonance energy zone to the thermal neutron energy zone,
R 2 n (u) -I 2 A basis function of the region;
I 2 -an indistinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy interval calculated in the step 5 is utilized to obtain a lower scattering source item from the distinguishable resonance energy zone to the thermal neutron energy zone,
R 3 n (u) -I 3 A basis function of the region;
I 3 -a distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
in a thermal neutron energy region, the microscopic cross section of nuclear reaction continuously and smoothly changes along with the energy of the incident neutrons, so that the corresponding neutron flux density also shows a trend of continuous and smooth change; according to the characteristics, a smooth basis function is selected, then the basis function expansion is carried out on the neutron angular flux density of the thermal energy region, and an equation about the neutron angular flux density in the energy phase space is converted into an equation about the neutron angular flux density expansion coefficient moment in the frequency domain space. The neutron transport equation format after the expansion of the basis functions of all the energy sections of the energy region is the formula (4), the neutron angular flux density coefficient moment and the neutron source strong expansion coefficient moment and the neutron angular flux density function of all the energy sections of the distinguishable resonance energy region are obtained by solving the equation,
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k nuclear density (cm) of the kth nuclear species at (r) -space r -3 )；
σ t,k Microscopic total cross section (cm) of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u 2 )；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth nuclide reacts with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and generates microscopic scattering cross sections (cm) of neutrons with logarithmic energy u, flight direction Ω% 2 )；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
R 2 n (u) -I 2 A basis function of the region;
R 3 n (u) -I 3 A basis function of the region;
R 4 n (u) -I 4 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
I 3 -a distinguishable resonance energy interval;
I 4 -thermal neutron energy interval;
step 7: returning to the step 5, forming up-scattering iteration, then checking the relative errors 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 that calculation converges if the relative errors do not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy interval; if the relative error exceeds the error limit, judging that the relative error is not converged, and continuing the processes from the step 5 to the step 6 until the set calculation iteration times are reached;
step 8: the obtained neutron angular flux expansion coefficient moment and the basis functions of selected areas in different energy intervals are utilized to obtain neutron angular flux density distribution of continuous energy in all the energy intervals, so that the description of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of nuclides, neutron beam intensity and irradiation environment parameters are quantitatively described.
Aiming at the characteristics of a nuclear reaction microcosmic section in a non-resonance energy region, an indistinguishable resonance energy region and a distinguishable resonance energy region, a targeted function expansion technology is provided, the net coupling characteristic of neutrons in an energy space is properly described, numerical dispersion is carried out on a neutron transport equation of continuous energy, the solution of neutron flux density distribution is converted into the solution of corresponding expansion coefficient moment, and the continuous distribution of physical quantities such as neutron flux density, nuclear reaction rate and the like along with the neutron energy is given. And further, the description of various nuclear reaction rates of various nuclides is realized, and data support is provided for further quantitatively describing the release of the fission energy, the consumption and production of the nuclides, the neutron beam intensity, the irradiation environment parameters and the like. The method has the outstanding advantages that: 1) Eliminating the multi-group database making link, avoiding the problems caused by the prior use of typical neutron energy spectrum, 2) eliminating the resonance self-shielding calculation link, and avoiding the self-shielding and mutual shielding effect correction problems in space, energy and nuclide constitution. 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 simulation can be simultaneously carried out under typical working conditions according to a design scheme, and the time cost is greatly reduced; on the other hand, the nuclear energy device has huge experimental analysis cost and possibly causes harm to experimental staff, so that the method can reduce the times of experimental analysis and lower the huge experimental cost on the premise of ensuring certain calculation accuracy. The design thought can be provided for the design of the related nuclear energy device, and quantitative data can be provided for further improving the performance of the nuclear energy device.
Claims (1)
1. A continuous energy definite theory neutron transport calculating method based on function expansion is characterized in that: the method comprises the following steps:
step 1: the energy region is divided into the following steps according to the microcosmic 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; wherein each energy interval can be divided into a plurality of energy sections;
step 2: different basis functions R are selected according to the need in different energy intervals n (u), n=0, 1,2, …, N, and functionally expanding the neutron angular flux density over different energy intervals to a form of the sum of products of unknown coefficients and different order basis functions, as follows:
phi (r, Ω, u) -neutron angular flux density in units of: cm -2 s -1 The method comprises the steps of carrying out a first treatment on the surface of the Where u= lnE, E is the outgoing neutron flight energy in units of: meV;
ψ n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;
base function R selected in different energy intervals n (u) can accurately describe the fluctuation change of neutron angular flux density along with energy in the energy interval; the larger the expansion order N of the basis function is, the more the depiction precision of different energy areas can be met;
step 3: selecting a smooth change basis function, only considering a self-scattering source item of an instinct, expanding the basis function of neutron angular flux density of a fast neutron energy zone, converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation format after the expansion of the energy zone function is a formula (1), and solving the equation to obtain neutron angular flux density coefficient moment and neutron angular flux density functions of each energy section of the fast neutron energy zone;
wherein:
omega-angle of exit neutrons in space;
angle of incidence neutrons in Ω' space;
r- -the spatial position;
logarithmic energy of the u-exit neutron energy;
logarithmic energy of u' -incident neutron energy;
N k the nuclear density of the kth species at (r) -space r in units of: cm -3 ；
σ t,k A microscopic total cross-section of nuclear reaction of (u) -kth species with neutrons having a logarithmic energy u in units of: cm 2 ；
σ s,k (u '. Fwdarw.u, Ω'. Fwdarw.Ω) -. Kth species react with scattering nuclei of neutrons with logarithmic energy u ', flight direction Ω' and produce microscopic scattering cross sections of neutrons with logarithmic energy u, flight direction Ω in units of: cm 2 ；
q n (r, Ω) -the neutron source strength to be solved for an unknown moment of expansion coefficient;
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
step 4: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source term from the fast neutron energy region to the indistinguishable resonance energy region,
R 1 n (u) -I 1 A basis function of the region;
I 1 -fast neutron energy interval;
I 2 -an indistinguishable resonance energy interval;
then, considering a self-scattering source item in an indistinguishable resonance energy interval, wherein an excitation curve of a microscopic section along with the change of the incident neutron energy in the indistinguishable resonance energy interval is provided with excessively dense resonance peaks, a determined value of the microscopic section of the appointed incident neutron energy cannot be determined experimentally, and only the probability density of the determined value in a certain value range can be given, which is called probability measure, and the probability measure is expressed in a probability table form; the method comprises the steps of using a basis function capable of considering energy distribution and simultaneously accurately measuring probability distribution, then performing basis function expansion on neutron angular flux density of an indistinguishable resonance energy region, converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation format after the energy region function expansion is formula (2), and solving the equation to obtain neutron angular flux density coefficient moment and neutron source strong expansion coefficient moment of each energy section of the indistinguishable resonance energy region and neutron angular flux density function;
wherein:
R 2 n (u) -I 2 A basis function of the region;
I 2 -an indistinguishable resonance energy interval;
step 5: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source term from the fast neutron energy region to the distinguishable resonance energy region,
I 3 -a 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 is utilized to obtain the lower scattering source item from the indistinguishable resonance energy zone to the distinguishable resonance energy zone,
in addition, the thermal neutron energy region is needed to reach the upper scattering source item of the discernible resonance energy region, the upper scattering source item needs to provide an initial value in the 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 4 A basis function of the region;
I 4 -thermal neutron energy interval;
using a basis function capable of accurately representing abrupt changes of neutron flux density, then performing basis function expansion on neutron angular flux density of a distinguishable resonance energy region, and converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space; the neutron transport equation format after the base function of each energy segment of the energy region is expanded is formula (3), and the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy segment of the distinguishable resonance energy region are obtained by solving the equation;
wherein:
R 3 n (u) -I 3 A basis function of the region;
step 6: the neutron angular flux density coefficient moment of each energy section of the fast neutron energy interval calculated in the step 3 is utilized to obtain the lower scattering source items from the fast neutron energy region to the thermal neutron energy region,
the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval calculated in the step 4 is utilized to obtain the lower scattering source item from the indistinguishable resonance energy zone to the thermal neutron energy zone,
obtaining a lower scattering source item from the distinguishable resonance energy region to the thermal neutron energy region by utilizing 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 neutron angular flux density of a thermal energy region, and converting an equation about neutron angular flux density in an energy phase space into an equation about neutron angular flux density expansion coefficient moment in a frequency domain space; the neutron transport equation format after the expansion of the basis functions of all the energy sections of the energy region is the formula (4), the neutron angular flux density coefficient moment and the neutron source strong expansion coefficient moment and the neutron angular flux density function of all the energy sections of the distinguishable resonance energy region are obtained by solving the equation,
step 7: returning to the step 5, forming up-scattering iteration, then checking the relative errors 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 that calculation converges if the relative errors do not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy interval; if the relative error exceeds the error limit, judging that the relative error is not converged, and continuing the processes from the step 5 to the step 6 until the set calculation iteration times are reached;
step 8: and obtaining neutron angular flux density distribution of continuous energy in all energy intervals by utilizing the obtained neutron angular flux expansion coefficient moment and the basis functions of selected areas in different energy intervals, so as to further realize the characterization of various nuclear reaction rates of various nuclides, and quantitatively describe the release of the fission energy, the consumption and production of the nuclides, the neutron beam intensity and irradiation environment parameters.
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