CN115048810A - Three-dimensional neutron transport equation calculation method and system based on linear source acceleration - Google Patents
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Abstract
The invention discloses a three-dimensional neutron transport equation calculation method and a three-dimensional neutron transport equation calculation system based on linear source acceleration, which relate to the technical field of nuclear reactor core design and are improved on the basis of a traditional flat source approximation method; the plane source approximation is eliminated theoretically, the number of grids of the plane source area is reduced, and the calculation efficiency is improved.
Description
Technical Field
The invention relates to the technical field of nuclear reactor core design, in particular to a linear source acceleration-based three-dimensional neutron transport equation calculation method and system.
Background
The reactor physical analysis calculation is used as the basis of the nuclear reactor system analysis calculation, and the reactor core reactivity and the full-reactor fine power distribution are obtained by solving a neutron transport equation. In order to rapidly develop the research and development of the advanced nuclear power reactor core, advanced high-precision reactor physical design software needs to be researched and developed. In order to simulate the core with a complex structure, research on a 'one-step' reactor physical calculation method based on an accurate physical model and a fine geometric modeling is widely carried out at home and abroad. The neutron angular flux of the three-dimensional neutron transport equation contains 7 dependent variables (3 dimensions in space, 2 dimensions in angle, 1 dimension in energy and 1 dimension in time), and accurate numerical simulation is very difficult.
The method for directly solving the three-dimensional neutron transport equation by the one-step method has the advantages of large calculation amount, high memory consumption and difficulty in realization under the existing calculation condition, so that a two-dimensional one-dimensional method is provided, the direct three-dimensional solution is converted into axial one-dimensional solution and radial two-dimensional solution respectively, and the coupling is performed through a leakage item, so that the calculation requirement of the one-step method is reduced. In the two-dimensional one-dimensional coupling method, flat source approximation is adopted, in order to obtain an accurate calculation result, a fine flat source region grid needs to be divided, the calculation efficiency is reduced, and the calculation requirement for solving a three-dimensional neutron transport equation by a one-step method is reduced by coupling through a leakage item. However, in the conventional two-dimensional one-dimensional coupling method, in order to obtain an accurate calculation result, a fine flat source area grid needs to be divided, so that the calculation efficiency is linearly reduced.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: in the traditional two-dimensional one-dimensional coupling method, in order to obtain an accurate calculation result, a fine flat source area grid needs to be divided, so that the calculation efficiency is linearly reduced; the invention aims to provide a three-dimensional neutron transport equation calculation method and system based on linear source acceleration, which are improved on the traditional plain form, carry out flux expansion based on the linear source form to calculate three-dimensional neutron flux and high-order standard flux moment, and solve the problem of low calculation efficiency caused by the fact that a plain source region grid is divided densely in the traditional two-dimensional one-dimensional method.
The invention is realized by the following technical scheme:
the invention provides a linear source acceleration-based three-dimensional neutron transport equation calculation method, which comprises the following steps of:
establishing a three-dimensional neutron transport equation;
converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
respectively solving the one-dimensional equation and the two-dimensional equation to obtain a characteristic value and a three-dimensional neutron flux of the reactor core; and when the two-dimensional equation is solved, flux expansion is carried out on the basis of a linear source form to calculate three-dimensional neutron flux and high-order standard flux moment.
The working principle of the scheme is as follows: in the traditional two-dimensional one-dimensional coupling method, in order to obtain an accurate calculation result, a fine flat source region grid needs to be divided, so that the calculation efficiency is linearly reduced, the invention aims to provide a three-dimensional neutron transport equation calculation method and a system based on linear source acceleration, the method is improved in the traditional plain form, the three-dimensional neutron flux and the high-order standard flux moment are calculated by performing flux expansion based on the linear source form, the high-order expansion is performed on the space through the linear source form, the high-order standard flux moment is obtained at the same time, and the approximation introduced by the flat source method is eliminated, so that the problem of low calculation efficiency caused by the division of the denser flat source region grid in the traditional two-dimensional one-dimensional method is solved; the scheme theoretically eliminates the flat source approximation introduced in the two-dimensional one-dimensional method, reduces the number of grids in the flat source region and improves the calculation efficiency.
The further optimization scheme is that the three-dimensional neutron transport equation is as follows:
wherein m represents an angle, g represents an energy group,. phi g,m (x, y, z) represents angular flux, x, y, z represent x, y, z coordinates of the location in space, ξ, respectively m Representing the cosine of the angle of incidence of the azimuth and the x-axis, sigma t,g (r) represents the total cross-section, η represents the amplitude sine, μ represents the polar cosine, Q g (x, y, z) represents the total source term.
The further optimization scheme is that the method for acquiring the one-dimensional equation and the two-dimensional equation comprises the following steps:
and (3) integrating the radial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a one-dimensional equation:
and (3) integrating the axial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a two-dimensional equation:
wherein psi g,m,i,j (z) angular flux, Q, of the z-th layer representing the radial (i, j) position angle m energy group g g,i,j (z) a one-dimensional total source term, Q, representing the radial (i, j) position g (x, y) represents a two-dimensional total source term,
The further optimization scheme is that the method for acquiring the characteristic value and the three-dimensional neutron flux of the reactor core comprises the following steps:
calculating fission sources and scattering sources;
on the basis of a fission source and a scattering source, respectively solving a one-dimensional equation and a two-dimensional equation to obtain radial flow, axial flow and radial flux;
performing CMFD iterative updating based on the radial flow, the axial flow and the radial flux to obtain a reactor core characteristic value and a three-dimensional neutron flux;
the CMFD iterative update process comprises:
calculating three-dimensional neutron average flux, flow coupling factors and homogenization cross sections according to the radial flow, the axial flow and the radial flux;
and updating the three-dimensional neutron flux and the characteristic value based on the CMFD characteristic value iteration result, and calculating a leakage item.
The further optimization scheme is that when the two-dimensional equation is solved, linear source item calculation is firstly carried out, then linear source characteristic line scanning is carried out, and finally the fine network standard flux and the high-order standard flux distance are calculated.
The further optimization scheme is that the method for acquiring the characteristic value and the three-dimensional neutron flux of the reactor core further comprises the following steps:
and after obtaining the three-dimensional neutron flux and the characteristic value through CMFD iterative updating, judging whether the current three-dimensional neutron flux and the characteristic value are both converged, if so, outputting the current three-dimensional neutron flux and the characteristic value, and otherwise, recalculating the fission source and the scattering source.
The specific two-dimensional equation solving method comprises the following steps:
writing a two-dimensional equation into a characteristic linear equation;
deducing an analytical solution of the angular flux and the first-order integral angular flux according to a characteristic linear equation;
and extracting a high-order standard flux moment from the analytic solutions of the angular flux and the first-order integral angular flux.
The further optimization scheme is that a two-dimensional neutron transport equation with an introduced axial leakage term is solved by a linear source characteristic line method, and the equation can be written into a characteristic line formula equation, wherein the characteristic linear equation is expressed as:
wherein,
is the angular flux in the i direction m of the flat source region, sigma t,i Is the total cross-section of the pipe,
is the total source term of the order 0,
is the 1 st order source term expansion moment, s i,m Is the characteristic line length, s i,m,k Is the length of the characteristic line segment k.
The further optimization scheme is that an analytic solution of the angular flux and the first-order integral angular flux is derived according to a characteristic linear equation, wherein the angular flux is expressed as:
the analytical solution for the first order integral angular flux is expressed as:
wherein,
is the incident angle flux, F 1 (Σ t,i s)、F 2 (Σ t,i s)、G 1 (Σ t,i s k )、G 2 (Σ t,i s k ) Are all high order coefficients.
The further optimization scheme is that the higher-order scalar quantity moment is represented by a higher-order coefficient:
F 1 (Σ t,i s)＝1-exp(-Σ t,i s)
F 2 (Σ t,i s)＝2[Σ t,i s-F 1 (Σ t,i s)]-Σ t,i s i,m,k F 1 (Σ t,i s)
wherein s is k The length of the characteristic line segment k is indicated and s the characteristic line length.
The scheme provides a linear source acceleration-based three-dimensional neutron transport equation calculation system, which is applied to the linear source acceleration-based three-dimensional neutron transport equation calculation method and comprises the following steps: the device comprises a construction module, a conversion module and a calculation module;
the construction module is used for establishing a three-dimensional neutron transport equation;
the conversion module is used for converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
the calculation module is used for solving the one-dimensional equation and the two-dimensional equation respectively to obtain a reactor core characteristic value and three-dimensional neutron flux; the calculation module is also used for calculating the three-dimensional neutron flux and the high-order scalar flux moment based on the linear source form for flux expansion when solving the two-dimensional equation.
The present solution provides a non-transitory computer readable storage medium having stored thereon computer instructions for execution by a processor to perform the steps described for implementing a linear source acceleration based three-dimensional neutron transport equation calculation method.
Compared with the prior art, the invention has the following advantages and beneficial effects:
the three-dimensional neutron transport equation calculation method and system based on linear source acceleration provided by the invention are improved on the traditional plain form, flux expansion is carried out on the basis of the linear source form to calculate three-dimensional neutron flux and high-order standard flux moment, high-order expansion is carried out on the space through the linear source form, and the high-order standard flux moment is obtained at the same time, so that approximation introduced by the plain source method is eliminated, and the problem of low calculation efficiency caused by the fact that a dense plain source area grid is divided in the traditional two-dimensional one-dimensional method is solved; the method theoretically eliminates the flat source approximation, reduces the number of grids in the flat source region and improves the calculation efficiency.
Drawings
In order to more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the drawings that are required in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and that those skilled in the art may also derive other related drawings based on these drawings without inventive effort. In the drawings:
FIG. 1 is a schematic flow chart of a method for calculating a three-dimensional neutron transport equation based on relaxation factors;
FIG. 2 is a schematic diagram of the two-dimensional/one-dimensional method basic principle;
FIG. 3 is a schematic diagram of a general flow of solving and calculating two-dimensional equations and one-dimensional equations;
fig. 4 is a schematic diagram of a two-dimensional equation scanning calculation solution flow.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to examples and accompanying drawings, and the exemplary embodiments and descriptions thereof are only used for explaining the present invention and are not meant to limit the present invention.
The three-dimensional neutron transport equation is directly solved by a one-step method, so that the calculation amount is large, the memory consumption is high, and the method is difficult to realize under the existing calculation condition. Therefore, a two-dimensional/one-dimensional method is provided, direct three-dimensional solution is converted into axial one-dimensional solution and radial two-dimensional solution respectively, and coupling is carried out through leakage items, so that the calculation requirement of directly solving a three-dimensional neutron transport equation by a one-step method is reduced; in the traditional two-dimensional one-dimensional coupling method, in order to obtain an accurate calculation result, a fine flat source area grid needs to be divided, so that the calculation efficiency is linearly reduced; the present invention provides the following embodiments to solve the above problems:
example 1
The embodiment provides a linear source acceleration-based three-dimensional neutron transport equation calculation method, as shown in fig. 1 and 2, including the steps of:
establishing a three-dimensional neutron transport equation;
converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
respectively solving the one-dimensional equation and the two-dimensional equation to obtain a reactor core characteristic value and three-dimensional neutron flux; and when the two-dimensional equation is solved, flux expansion is carried out on the basis of a linear source form to calculate three-dimensional neutron flux and high-order standard flux moment.
The three-dimensional neutron transport equation is:
wherein m represents an angle, g represents an energy group,. phi g,m (x, y, z) represents angular flux, x, y, z represent x, y, z coordinates of the location in space, ξ, respectively m Representing the cosine of the angle of incidence of the azimuth and the x-axis, sigma t,g (r) represents the total cross-section, η represents the amplitude sine, μ represents the polar cosine, Q g (x, y, z) represents the total source term.
The one-dimensional equation and the two-dimensional equation are obtained by the following method:
and (3) integrating the radial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a one-dimensional equation:
and (3) integrating the axial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a two-dimensional equation:
wherein psi g,m,i,j (z) angular flux, Q, of the z-th layer representing the radial (i, j) position angle m energy group g g,i,j (z) a one-dimensional total source term, Q, representing the radial (i, j) position g (x, y) represents a two-dimensional total source term,
The method for acquiring the characteristic value and the three-dimensional neutron flux of the reactor core comprises the following steps:
calculating fission sources and scattering sources;
on the basis of a fission source and a scattering source, respectively solving a one-dimensional equation and a two-dimensional equation to obtain radial flow, axial flow and radial flux;
performing CMFD iterative updating based on the radial flow, the axial flow and the radial flux to obtain a reactor core characteristic value and a three-dimensional neutron flux;
the CMFD iterative update process comprises:
calculating three-dimensional neutron average flux, flow coupling factors and homogenization cross sections according to the radial flow, the axial flow and the radial flux;
and updating the three-dimensional neutron flux and the characteristic value based on the CMFD characteristic value iteration result, and calculating a leakage item.
As shown in fig. 4, the two-dimensional equation solving method includes the steps of: and (3) performing linear source item calculation, performing linear source characteristic line scanning, and finally calculating the fine network standard flux and the high-order standard flux distance.
Specifically, a two-dimensional equation is written into a characteristic linear equation;
deducing an analytical solution of the angular flux and the first-order integral angular flux according to a characteristic linear equation;
and extracting a high-order standard flux moment from the analytic solutions of the angular flux and the first-order integral angular flux.
The characteristic linear equation is expressed as:
wherein,
is the angular flux in the i direction m of the flat source region, sigma t,i Is the total cross-section of the pipe,
is the total source term of the order 0,
is the 1 st order source term expansion moment, s i,m Is the characteristic line length, s i,m,k Is the length of the characteristic line segment k.
The angular flux is expressed as:
the analytical solution for the first order integral angular flux is expressed as:
wherein,
is the incident angle flux, F 1 (Σ t,i s)、F 2 (Σ t,i s)、G 1 (Σ t,i s k )、G 2 (Σ t,i s k ) Are all high order coefficients.
The higher order scalar flux moment is represented by a higher order coefficient:
F 1 (Σ t,i s)＝1-exp(-Σ t,i s)
F 2 (Σ t,i s)＝2[Σ t,i s-F 1 (Σ t,i s)]-Σ t,i s i,m,k F 1 (Σ t,i s)
wherein s is k The length of the characteristic line segment k is indicated and s the characteristic line length.
Example 2
The embodiment provides a linear source acceleration-based three-dimensional neutron transport equation calculation system, which is applied to the method in the previous embodiment, and is characterized by comprising the following steps: the device comprises a construction module, a conversion module and a calculation module;
the construction module is used for establishing a three-dimensional neutron transport equation;
the conversion module is used for converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
the calculation module is used for solving the one-dimensional equation and the two-dimensional equation respectively to obtain a reactor core characteristic value and three-dimensional neutron flux; the calculation module is also used for calculating the three-dimensional neutron flux and the high-order scalar flux moment based on the linear source form for flux expansion when solving the two-dimensional equation.
Example 3
The present embodiments provide a non-transitory computer readable storage medium having stored thereon computer instructions that, when executed by a processor, implement the steps of the method of embodiment 1.
The processor performs the steps further comprising:
(1) before iterative computation, geometric modeling is carried out, a characteristic line is generated, and linear source related parameters are computed;
(2) calculating fission source and scattering source items;
(3) and respectively carrying out one-dimensional equation solution and two-dimensional equation solution to obtain axial flow, radial flow and radial flux. The method comprises the following steps of solving a two-dimensional equation, wherein the two-dimensional equation is based on a linear source method, namely linear source item calculation, characteristic line scanning based on the linear source method, and standard flux and high-order standard flux moment calculation are carried out;
(4) carrying out coarse net effective differential Calculation (CMFD) according to two-dimensional and one-dimensional calculation results;
(5) and (5) judging whether the characteristic value and the crack rate are converged, finishing the calculation if the characteristic value and the crack rate are converged, and returning to the step (2) if the characteristic value and the crack rate are not converged.
In the step (3), the two-dimensional calculation by using the linear source method is the key of the embodiment. The method is improved on the basis of the traditional flat source approximation method, the high-order scalar flux moment is additionally calculated in the processes of source item calculation, characteristic line scanning, flux calculation and the like, high-order expansion is carried out in space, approximation introduced by the flat source method is eliminated theoretically, and therefore accurate calculation results can be obtained by adopting fewer flat source area grids, and calculation efficiency is improved.
The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are merely exemplary embodiments of the present invention, and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (10)
1. The linear source acceleration-based three-dimensional neutron transport equation calculation method is characterized by comprising the following steps of:
establishing a three-dimensional neutron transport equation;
converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
respectively solving the one-dimensional equation and the two-dimensional equation to obtain a reactor core characteristic value and three-dimensional neutron flux; and when the two-dimensional equation is solved, flux expansion is carried out on the basis of a linear source form to calculate three-dimensional neutron flux and high-order standard flux moment.
2. The linear source acceleration based three-dimensional neutron transport equation calculation method of claim 1, wherein the three-dimensional neutron transport equation is:
wherein m represents an angle, g represents an energy group,. phi g,m (x, y, z) represents angular flux, x, y, z represent x, y, z coordinates of the location in space, ξ, respectively m Representing the cosine of the angle of incidence of the azimuth and the x-axis, sigma t,g (r) represents the total cross-section, η represents the amplitude sine, μ represents the polar cosine, Q g (x, y, z) represents the total source term.
3. The linear source acceleration-based three-dimensional neutron transport equation calculation method according to claim 2, wherein the one-dimensional equation and the two-dimensional equation are obtained by:
and (3) integrating the radial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a one-dimensional equation:
and (3) integrating the axial direction in the area of each rod of each layer by using a three-dimensional neutron transport equation to obtain a two-dimensional equation:
wherein psi g,m,i,j (z) angular flux, Q, of the z-th layer representing the radial (i, j) position angle m energy group g g,i,j (z) a one-dimensional total source term, Q, representing the radial (i, j) position g (x, y) represents a two-dimensional total source term,
4. The linear source acceleration-based three-dimensional neutron transport equation calculation method according to claim 1, wherein the reactor core characteristic value and three-dimensional neutron flux acquisition method comprises the steps of:
calculating fission sources and scattering sources;
on the basis of a fission source and a scattering source, respectively solving a one-dimensional equation and a two-dimensional equation to obtain radial flow, axial flow and radial flux;
performing CMFD iterative updating based on the radial flow, the axial flow and the radial flux to obtain a reactor core characteristic value and a three-dimensional neutron flux;
the CMFD iterative update process comprises:
calculating three-dimensional neutron average flux, flow coupling factors and homogenization cross sections according to the radial flow, the axial flow and the radial flux;
and updating the three-dimensional neutron flux and the characteristic value based on the CMFD characteristic value iteration result, and calculating a leakage item.
5. The linear source acceleration-based three-dimensional neutron transport equation calculation method according to claim 3, wherein the two-dimensional equation solving method comprises the steps of:
writing a two-dimensional equation into a characteristic linear equation;
deducing an analytical solution of the angular flux and the first-order integral angular flux according to a characteristic linear equation;
and extracting a high-order standard flux moment from the analytic solutions of the angular flux and the first-order integral angular flux.
6. The linear source acceleration-based three-dimensional neutron transport equation calculation method of claim 5, wherein the characteristic linear equation is expressed as:
wherein,
is the angular flux in the i direction m of the flat source region, sigma t,i Is the total cross-section of the pipe,
is the total source term of the order 0,
is the unfolding moment of the source term of order 1, s i,m Is the characteristic line length, s i,m,k Is the length of the characteristic line segment k.
7. The linear source acceleration-based three-dimensional neutron transport equation calculation method of claim 6, wherein the angular flux is expressed as:
the analytical solution for the first order integral angular flux is expressed as:
wherein,
is the incident angle flux, F 1 (Σ t,i s)、F 2 (Σ t,i s)、G 1 (Σ t,i s k )、G 2 (Σ t,i s k ) Are all high order coefficients.
8. The linear-source-acceleration-based three-dimensional neutron transport equation calculation method of claim 7, wherein the higher-order scalar moments are represented by higher-order coefficients:
F 1 (Σ t,i s)＝1-exp(-Σ t,i s)
F 2 (Σ t,i s)＝2[Σ t,i s-F 1 (Σ t,i s)]-Σ t,i s i,m,k F 1 (Σ t,i s)
wherein s is k The length of the characteristic line segment k is indicated and s the characteristic line length.
9. The linear source acceleration-based three-dimensional neutron transport equation calculation system is applied to the method of any one of claims 1 to 8, and is characterized by comprising the following steps of: the device comprises a construction module, a conversion module and a calculation module;
the construction module is used for establishing a three-dimensional neutron transport equation;
the conversion module is used for converting a three-dimensional neutron transport equation into a one-dimensional equation and a two-dimensional equation;
the calculation module is used for solving the one-dimensional equation and the two-dimensional equation respectively to obtain a reactor core characteristic value and three-dimensional neutron flux; the calculation module is also used for calculating the three-dimensional neutron flux and the high-order scalar flux moment based on the linear source form for flux expansion when solving the two-dimensional equation.
10. A non-transitory computer readable storage medium having stored thereon computer instructions, characterized in that the instructions, when executed by a processor, implement the steps of the method of any one of claims 1-8.
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| CN112632822A (en) * | 2020-12-21 | 2021-04-09 | 中国核动力研究设计院 | Nuclear reactor neutron flux obtaining method and device based on three-dimensional leakage item segmentation |
| CN113672849A (en) * | 2021-08-26 | 2021-11-19 | 中国核动力研究设计院 | One-step transportation calculation method and system based on axial flux expansion |
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