Variational slab process method, system, and medium for complex stack type core determination

Abstract

This invention belongs to the field of nuclear reactor core calculation technology, and discloses a variational nodal block processing method, system, and medium for determining the core of complex reactor types. The variational nodal block processing method for determining the core of complex reactor types includes: establishing and solving the three-dimensional quasi-diffusion equation; calculating the Eddington factor; and analyzing the calculation accuracy and efficiency of the quasi-diffusion equation. This invention, through the derivation, calculation, and verification of the quasi-diffusion equation based on the variational nodal block method for solving hexagonal geometry, establishes a practical, efficient, and accurate method for calculating reactor core neutron flux. This method can be used to solve complex or simple reactor types with various hexagonal geometries, effectively overcoming the accuracy problems of traditional diffusion equation calculations and the efficiency problems of transport equations, and has significant implications and practical engineering value.

Description

Technical Field

[0001] This invention belongs to the field of nuclear reactor core calculation technology, and particularly relates to a method, system, and medium for handling variable segmentation blocks in the determination of cores for complex reactor types. The variable segmentation block method is used to solve the quasi-diffusion equations under hexagonal geometry. Background Technology

[0002] Currently, with the development of nuclear energy, various advanced reactor types are emerging, and complex reactor types based on hexagonal components are becoming common. These reactor types typically have more advanced fuel types, higher enrichment, more inhomogeneous core arrangements, complex component and core geometry, strong neutron anisotropy, and drastic local neutron flux variations. In these cases, the traditional diffusion approximation assumptions used for core calculations are no longer applicable. While transport calculations can guarantee good accuracy, their computational efficiency is too low. Based on these issues, this paper proposes a variable segmentation block processing method for determining the core of complex reactor types. The main difference between quasi-diffusion and traditional diffusion equations is the addition of a transport correction term, the Eddington factor, to the neutron diffusion equation to describe the angular distribution effect of neutron flux. One of the basic assumptions in the traditional diffusion theory is that the angular flux of neutrons is a first-order function of angle. Quasi-diffusion theory does not adopt this assumption; instead, it uses the Eddington factor to represent the integral term of the product of the neutron angular flux and the angle vector. Therefore, quasi-diffusion theory offers better accuracy compared to traditional diffusion theory. Furthermore, the form of the quasi-diffusion equations is similar to that of traditional diffusion, and their numerical discretization is consistent, making their computational speed comparable to traditional diffusion methods. The variational nodal method is used primarily to expand the relevant variables within the three-dimensional nodal by utilizing orthogonal basis functions, avoiding singular terms caused by lateral leakage when using traditional lateral integration methods to handle hexagonal problems. Therefore, the quasi-diffusion variational nodal method based on hexagonal components is of significant importance and practical engineering value for the neutronics analysis and design of novel reactors (space reactors, fast neutron reactors, dynamic reactors, etc.) with complex energy spectra, core arrangements, and strong anisotropy.

[0003] Hall used the modified Serpent to generate the component homogenization cross section and Eddington factor, briefly described the numerical calculation method of the quasi-diffusion equation in the one-dimensional case, and derived the form of the nodal discontinuity factor under the quasi-diffusion theory. Finally, the verification analysis was performed using a one-dimensional RBWR component (see...). Figure 5This paper utilizes Serpent to generate homogenized cross sections and Eddington factors for reactor cores. Based on the definition of the Eddington factor, the quasi-diffusion equation is derived from the transport equation, and a solution for the one-dimensional quasi-diffusion equation is provided. Furthermore, the calculation method for discontinuity factors after introducing the Eddington factor is extended. The numerical solution of the quasi-diffusion equation for a one-dimensional RBWR reactor core is compared and analyzed with the Pn equation. However, existing techniques only provide numerical solutions for the one-dimensional quasi-diffusion equation, without addressing solutions for complex cores such as two-dimensional and three-dimensional cores. Secondly, the specific method for solving the quasi-diffusion equation is the nodal expansion method, which is mainly applicable to rectangular nodules and has limitations for advanced reactor types with hexagonal components. Therefore, there is an urgent need to design a new variational nodal processing method, system, and medium for determining the core of complex reactor types.

[0004] Based on the above analysis, the problems and shortcomings of the existing technology are as follows:

[0005] (1) Existing technologies only provide numerical solutions for the one-dimensional solution of the quasi-diffusion equation, without providing solutions for complex cores such as two-dimensional and three-dimensional cores.

[0006] (2) In the prior art, the specific method for solving the quasi-diffusion equation is the block expansion method, which is mainly for rectangular blocks and has limitations for advanced stacking of hexagonal components.

[0007] (3) Existing methods have singular terms in the second-order tensor form of the diffusion term of the quasi-diffusion equation under hexagonal nodules.

[0008] (3) For complex hexagonal cores with strong anisotropy, existing technologies have limitations in ensuring high computational accuracy while maintaining high computational efficiency. Summary of the Invention

[0009] To address the problems existing in the prior art, the present invention provides a method, system, and medium for processing variable segment blocks for determining the core of complex reactor types.

[0010] This invention is implemented as follows: a method for processing variable segment blocks for determining the core of complex reactor types, the method comprising the following steps:

[0011] Step 1: Establish the Quasi-diffusion equations and corresponding boundary conditions under hexagonal geometry;

[0012] Step 2: Transform the leakage term of the second-order tensor using the variable segmentation block method to establish a variable segmentation block numerical calculation method for Quasi-diffusion under hexagonal geometry;

[0013] Step 3: The Monte Carlo program Serpent calculates the Eddington factor.

[0014] Step four: Analyze the computational accuracy and efficiency of the quasi-diffusion equation.

[0015] Furthermore, the establishment of the Quasi-diffusion equation and corresponding boundary conditions under hexagonal geometry in step one includes:

[0016] Similar to the derivation of the P1 equation, the neutron transport equation for time-independent continuous energy is integrated over Ω∈[0, 4π] and multiplied by Ω and integrated over the full angular space to obtain the neutron standard flux equation and the neutron flux equation, respectively. Unlike the derivation of the P1 equation, the derivation of this neutron flux equation does not introduce the assumption of a first-order approximation of the neutron angular flux density over the angular variable, and the Eddington factor is defined.

[0017] Furthermore, in step two, the variational block method is used to transform the leakage term of the second-order tensor, establishing a variational block numerical calculation method for Quasi-diffusion under hexagonal geometry, including:

[0018] According to Galerkin's variational principle, the Quasi-diffusion equation establishes a global functional F over the entire solution domain, which consists of several blocks, and its boundaries.

[0019] Furthermore, the calculation of the Eddington factor by the Monte Carlo program Serpent in step three includes:

[0020] The Eddington factor is treated as a small-group homogenization parameter similar to other parameters including the fission cross section and scattering cross section. The Monte Carlo program SERPENT source code is modified to enable the calculation of the Eddington factor. First, the flight direction Ω of each particle in the SERPENT source code is used as the basis for this calculation. u Ω v Ω w And the location of the neutron angular flux variable along this direction, by modifying the program source code to statistically analyze the following variables in 4π space.

[0021]

[0022] The standard neutron flux φ can be calculated in the original program, so the Eddington factor is finally calculated using the following formula.

[0023]

[0024] This invention proposes to solve the Quasi-diffusion equation based on a two-step core computation method. The conventional minority homogenization section and the Eddington factor, a key parameter in the Quasi-diffusion equation, are calculated using the Monte Carlo program SERPENT-2.1.26. Since the SERPENT program itself has a conventional minority homogenization section such as ∑... a ,∑ s ,∑ f It has the function to calculate parameters such as Eddington factor, but not Eddington factor. This invention formulates the following through the above method: Figure 6 The SERPENT program modification scheme is shown.

[0025] First, read the SERPENT source code and refer to the existing calculation process for the conventional homogenization cross-section variables. Taking the absorbing cross-section as an example, the variable in the code is defined as GCU_INF_ABS, which first needs to be defined in the header file location.h. Similarly, the variables related to the Eddington factor also need to be defined here, such as... Figure 7 As shown, GCU_INF_FLX_XX represents GCU_INF_FLX_XY indicates And so on.

[0026] Similar to the calculation of the absorption cross section, the quantities in the Eddington factor equation are calculated in the scoregc.h file: wgt*flx*xy, where wgt is the weight, flx is the neutron angular flux, and xy is the product of the angle between the angular flux direction and the x-axis and the y-axis. The result of this calculation is the numerator of the Eddington factor equation, and the denominator is calculated as wgt*flx.

[0027] To obtain the Eddington factor under an infinite energy spectrum, first add the microscopic components calculated above and store them in the Eddington factor calculation components, such as... Figure 9 middle.

[0028] Finally, the corresponding Eddington factor is output in the *.m file.

[0029] Furthermore, step four, the analysis of the computational accuracy and efficiency of this technology, includes:

[0030] To verify the computational accuracy and efficiency of quasi-diffusion theory under hexagonal component cores, advanced reactors, including RBWR and fast reactor benchmarks, were used to perform computational analysis on the benchmarks. Through keff and power density distribution error calculations, a comparative analysis of the quasi-diffusion method with traditional diffusion and transport methods was conducted in terms of computational accuracy and efficiency.

[0031] Another object of the present invention is to provide a variational nodal block processing system for hexagonal geometry equations applying the aforementioned variational nodal block processing method for determining complex reactor core configurations, the variational nodal block processing system for hexagonal geometry equations comprising:

[0032] The equation establishment and solution module is used to establish and solve three-dimensional quasi-diffusion equations.

[0033] The Eddington factor calculation module is used to perform Eddington factor calculations.

[0034] The computational accuracy and efficiency analysis module is used to analyze the computational accuracy and efficiency of the quasi-diffusion equation.

[0035] Another object of the present invention is to provide a computer device comprising a memory and a processor, the memory storing a computer program, which, when executed by the processor, causes the processor to perform the following steps:

[0036] The three-dimensional quasi-diffusion equations were established and solved; the Eddington factor was calculated; and the computational accuracy and efficiency of the quasi-diffusion equations were analyzed.

[0037] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the following steps:

[0038] The three-dimensional quasi-diffusion equations were established and solved; the Eddington factor was calculated; and the computational accuracy and efficiency of the quasi-diffusion equations were analyzed.

[0039] Another object of the present invention is to provide a computer program product stored on a computer-readable medium, comprising a computer-readable program that, when executed on an electronic device, provides a user input interface for applying the variational block processing system of the equations under the hexagonal geometry.

[0040] Another object of the present invention is to provide a computer-readable storage medium storing instructions that, when executed on a computer, cause the computer to apply the variational block processing system of the equations under the hexagonal geometry.

[0041] Another objective of this invention is to provide an information data processing terminal for implementing the variational segment block processing system for equations under hexagonal geometry.

[0042] Based on the above technical solutions and the technical problems solved, please analyze the advantages and positive effects of the technical solution to be protected by this invention from the following aspects:

[0043] First, addressing the technical problems existing in the prior art and the difficulty in solving them, this paper closely analyzes, in conjunction with the technical solution to be protected by this invention and the results and data obtained during the research and development process, how the technical solution of this invention solves the technical problems, and the inventive technical effects brought about by solving these problems. The specific description is as follows:

[0044] In light of the development of advanced reactor types based on hexagonal components, and considering the potential for uneven fuel distribution, complex neutron spectra, and strong neutron anisotropy, which leads to low accuracy in traditional diffusion calculations and low efficiency in transport calculations, this invention develops a novel, efficient, and accurate three-dimensional reactor core neutron calculation method based on the Eddington factor and quasi-diffusion. This method is used to solve the neutronics calculation problems of various reactor cores under hexagonal meshes.

[0045] For novel reactors with hexagonal components exhibiting complex neutron spectra and strong neutron anisotropy, this invention establishes a high-precision, computationally efficient core neutronics calculation method based on the variational segmentation block method to solve quasi-diffusion under hexagonal geometry. This is an innovation in core neutronics calculation methods, overcoming the accuracy problems of diffusion equations and the efficiency problems of transport equations, and has significant implications and practical engineering value.

[0046] Other methods can be used to solve for the Eddington factor. For example, the component program based on the deterministic MOC method can also be modified to calculate the Eddington factor, and then combined with some minority group homogenization parameters to achieve the purpose of solving the quasi-diffusion equation.

[0047] This invention provides a method for calculating the Eddington factor using the Monte Carlo method and for numerically solving the quasi-diffusion equation under hexagonal geometry. The intended protection of this invention lies in solving the quasi-diffusion equation under hexagonal geometry using the variational nodal block method. Furthermore, the derivation and calculation of the Eddington factor are also aspects protected by this invention.

[0048] The innovative aspects of this invention compared to existing technologies are as follows: First, the calculation method for solving the Eddington factor is innovative; second, currently, the use of quasi-diffusion equations to solve core problems both domestically and internationally mainly targets 1-dimensional or 2-dimensional problems. Therefore, using the variational segmentation block method to solve the 3-dimensional quasi-diffusion equations under hexagonal components is also one of the innovative aspects of this invention.

[0049] The inventive aspect of this invention is achieved through the following principles:

[0050] (1) In view of the Eddington factor, a key parameter in the solution process of the quasi-diffusion equation and considering the complexity of the neutron energy spectrum in the reactor core, a "two-step method" is adopted. The source code of the Monte Carlo program Serpent is modified to enable it to calculate the Eddington factor.

[0051] (2) When using the Eddington factor expression, the first-order angular flux approximation is introduced, and finally the three-dimensional quasi-diffusion equation is established.

[0052] (3) Drawing on the current mature numerical solution algorithm for traditional diffusion equations—the variational block method—we study its method for handling the leakage term of the quasi-diffusion equation and establish a numerical solution model for the three-dimensional quasi-diffusion equation.

[0053] When solving reactor core problems, existing technologies typically use traditional diffusion theory. However, diffusion theory has significant accuracy deviations for new reactor types with complex energy spectra, geometric structures, and strong neutron flux anisotropy. On the other hand, using transport theory to solve the problem with high accuracy is difficult to guarantee in terms of computational efficiency. Therefore, using the Eddington factor to solve the quasi-diffusion theory while considering both efficiency and accuracy is of great significance for the development and design of reactors.

[0054] Second, considering the technical solution as a whole or from a product perspective, the technical effects and advantages of the technical solution to be protected by this invention are specifically described as follows:

[0055] This invention derives, calculates, and verifies a practical, efficient, and accurate method for calculating neutron flux in reactor cores by solving the quasi-diffusion equation under hexagonal geometry using the variational segmentation method. This method can be used to solve complex or simple reactor configurations under various hexagonal geometries, effectively overcoming the accuracy problems of traditional diffusion equation calculations and the efficiency problems of transport equations. It has significant implications and practical engineering value.

[0056] Third, as supplementary evidence of the inventive step of the claims of this invention, it is also reflected in the following important aspects:

[0057] The technical solution of this invention fills a technological gap in the industry both domestically and internationally: Currently, the main methods for solving reactor core calculations both domestically and internationally are based on traditional diffusion theory for numerical solutions. However, with the continuous development and upgrading of reactors, more and more new and complex reactor cores are being introduced. These new reactor types often use more advanced fuel types and more non-uniform fuel arrangements, resulting in stronger neutron anisotropy. The traditional diffusion approximation assumptions are no longer applicable, and the calculation efficiency using transport equations is too low. Therefore, finding a reactor core calculation method that can guarantee both efficiency and accuracy has always been a key research goal both domestically and internationally. The Quasi-diffusion equation was proposed in 1964, but research on it both domestically and internationally has mainly focused on one-dimensional and two-dimensional reactor core calculations, and mainly on rectangular geometry. The technology for solving hexagonal three-dimensional reactor cores remains a blank.

[0058] On the one hand, this invention constructs the Quasi-diffusion equation for a three-dimensional reactor core based on the Eddington factor, and innovatively uses the variational segmentation block method to perform development calculations on hexagonal component reactor cores. This avoids the lateral leakage and singular terms generated by traditional lateral integration techniques when dealing with hexagonal Quasi-diffusion equations. Furthermore, it develops a reactor core calculation program that balances accuracy and efficiency, and verifies the accuracy and efficiency of its calculation results through some benchmark cases. To a certain extent, this fills a gap in the domestic and international three-dimensional hexagonal reactor core calculation and provides certain reference value for the development of reactor core calculation and other related technologies.

[0059] On the other hand, this invention, based on the Monte Carlo program Serpent, treats the Eddington factor as a similar minority group homogenization parameter to other parameters including fission cross-section and scattering cross-section. It employs the homogenization cross-section calculation concept from the "two-step method" of core computation, using component calculations to obtain the angular flux density distribution, and then calculating the Eddington factor in the homogenized region. By modifying the Serpent source code to enable it to calculate the Eddington factor, this invention also provides a computational approach for subsequent calculations of the Eddington factor. Attached Figure Description

[0060] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments of the present invention will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0061] Figure 1 This is a flowchart of the variable segment block processing method for determining the core of a complex reactor type, provided in an embodiment of the present invention.

[0062] Figure 2This is a schematic diagram of the variable segment block processing method for determining the core of complex reactor types provided in this embodiment of the invention;

[0063] Figure 3 This is a structural block diagram of the variational segmentation block processing system for equations under hexagonal geometry provided in an embodiment of the present invention;

[0064] Figure 4 This is a schematic diagram of a hexagonal block provided in an embodiment of the present invention;

[0065] Figure 5 This is a flowchart of the prior art provided in the embodiments of the present invention;

[0066] Figure 6 This is a schematic diagram of the SERPENT program modification scheme provided in the embodiments of the present invention;

[0067] Figure 7 This is a schematic diagram illustrating the definition of Eddington factor-related quantities in the Location.h header file provided in this embodiment of the invention;

[0068] Figure 8 This is a schematic diagram of the calculation of the Eddington factor correlation quantity in the Scoringc.c file provided in this embodiment of the invention;

[0069] Figure 9 This is a schematic diagram of the calculation of the Eddington factor components of the infinite energy spectrum provided in an embodiment of the present invention;

[0070] Figure 10 This is a schematic diagram of the output of the Eddington factor provided in an embodiment of the present invention;

[0071] Figure 11 This is provided by the embodiments of the present invention. Figure 11 Cross-sectional view of the spherical source;

[0072] Figure 12 This is a schematic diagram of the 1D RBWR problem model provided in an embodiment of the present invention;

[0073] Figure 13 This is a schematic diagram of the neutron flux error in the first group of nodal blocks provided in an embodiment of the present invention;

[0074] Figure 14 This is a schematic diagram of the neutron flux error in the fourth group of nodal blocks provided in an embodiment of the present invention;

[0075] Figure 15 This is a schematic diagram of the neutron flux error in the 8th group of nodal blocks provided in an embodiment of the present invention;

[0076] Figure 16 This is a schematic diagram of the neutron flux error in the 12th group of nodal blocks provided in an embodiment of the present invention;

[0077] Figure 17This is a schematic diagram of the neutron flux error in the first group of nodal blocks provided in an embodiment of the present invention;

[0078] Figure 18 This is a schematic diagram of the neutron flux error in the fourth group of nodal blocks provided in an embodiment of the present invention;

[0079] Figure 19 This is a schematic diagram of the neutron flux error in the 8th group of nodal blocks provided in an embodiment of the present invention;

[0080] Figure 20 This is a schematic diagram of the neutron flux error in the 12th group of nodal blocks provided in an embodiment of the present invention;

[0081] Figure 21 These are the radial distribution and radial distribution diagram of BN-600 provided in the embodiments of the present invention;

[0082] Figure 22 This is a schematic diagram of the radial arrangement of the virtual core provided in an embodiment of the present invention;

[0083] Figure 23 This is a schematic diagram of the SERPENT calculation area (the area enclosed by the red line) provided in an embodiment of the present invention;

[0084] Figure 24 This is a schematic diagram of the neutron flux error in the first group of nodal blocks provided in an embodiment of the present invention;

[0085] Figure 25 This is a schematic diagram of the neutron flux error in the fourth group of nodal blocks provided in an embodiment of the present invention;

[0086] Figure 26 This is a schematic diagram of the neutron flux error in the 8th group of nodal blocks provided in an embodiment of the present invention;

[0087] Figure 27 This is a schematic diagram of the neutron flux error in the 11th group of nodal blocks provided in an embodiment of the present invention;

[0088] Figure 28 This is a radial schematic diagram of a reactor core with LEZ and SHR fuel types in the BN600 reference reactor provided in an embodiment of the present invention;

[0089] Figure 29 This is a schematic diagram of component numbering provided in an embodiment of the present invention.

[0090] In the diagram: 1. Equation establishment and solution module; 2. Eddington factor calculation module; 3. Calculation accuracy and efficiency analysis module. Detailed Implementation

[0091] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0092] To address the problems existing in the prior art, the present invention provides a method, system, and medium for processing variable segment blocks for determining the core of complex reactor types. The present invention will be described in detail below with reference to the accompanying drawings.

[0093] I. Explanatory and Illustrative Embodiments. To enable those skilled in the art to fully understand how the present invention is specifically implemented, this section provides an explanatory and illustrative description of the embodiments described in the claims.

[0094] like Figure 1 As shown, the variable segment block processing method for determining the core of a complex reactor type provided in this embodiment of the invention includes the following steps:

[0095] S101, establish the three-dimensional quasi-diffusion equation under hexagonal geometry;

[0096] S102, Solving the quasi-diffusion equations in hexagonal geometry using the variable segmentation block method;

[0097] S103, perform Eddington factor calculation;

[0098] S104 performs calculation accuracy and efficiency analysis.

[0099] The schematic diagram of the variable segmentation block processing method for determining the core of complex reactor types provided in this embodiment of the invention is as follows: Figure 2 As shown.

[0100] like Figure 3 As shown, the variational block processing system for equations under hexagonal geometry provided in this embodiment of the invention includes:

[0101] Equation Establishment and Solution Module 1 is used to establish and solve three-dimensional quasi-diffusion equations;

[0102] Eddington Factor Calculation Module 2 is used to perform Eddington factor calculation;

[0103] Module 3, which analyzes the accuracy and efficiency of calculations for the quasi-diffusion equations, is used to perform such analyses. 1. Summary of the Invention

[0105] In light of the development of advanced reactor types based on hexagonal components, and considering the potential for uneven fuel distribution, complex neutron spectra, and strong neutron anisotropy, which leads to low accuracy in traditional diffusion calculations and low efficiency in transport calculations, this invention develops a novel, efficient, and accurate three-dimensional reactor core neutron calculation method based on the Eddington factor and quasi-diffusion. This method is used to solve the neutronics calculation problems of various reactor cores under hexagonal meshes.

[0106] like Figure 2 As shown, the variable segment block processing method for determining the core of a complex reactor type provided in this embodiment of the invention includes the following steps:

[0107] 1) Establishing the three-dimensional quasi-diffusion equations under hexagonal geometry.

[0108] Similar to the derivation of the P1 equation, integrating the time-independent neutron transport equation over Ω∈[0, 4π] and multiplying it by Ω and integrating over the entire angular space yields the neutron scalar flux equation and the neutron flux equation, respectively. It is worth noting that, unlike the derivation of the P1 equation, this derivation of the neutron flux equation does not introduce the assumption of a first-order approximation of the neutron angular flux density over the angular variable, and defines the Eddington factor as follows:

[0109]

[0110]

[0111] Where Ω represents the angle; Ω u Ω v Let ψ(r, E, Ω) represent the angles in each direction, φ(r, E) represent the neutron angular flux density, and Φ(r, E) represent the neutron flux density. u (r, E) represents the energy component in the u direction. These represent the unit vectors in each direction.

[0112] The expression for neutron flow is as follows:

[0113]

[0114] Where ∑ tr (r, E) is the neutron transport cross section, and J(r, E) is the neutron flux density.

[0115] Finally, the quasi-diffusion equation is obtained:

[0116]

[0117] Where, k eff χ(E) is the effective multiplication factor, v∑ is the fission energy spectrum. f(r, E′) is the neutron production cross section, ∑ s0 (r, E′→E) represents the scattering cross section from energy group E′ to energy group E. ∑ t (r, E) is the sum of the absorption cross section and the scattering cross section, used to calculate the consumption.

[0118] The continuity condition in the quasi-diffusion approximation equation is based on the angular flux Ψ g (r, E) is a continuous function of r at the interface between the two media, and can be approximated by the following set of conditions:

[0119] ∫ Ω n·ΩΨ g (r, Ω)Y n，m (Ω) (4)

[0120] Where n = 0, 1, ..., N is continuous at the interface, Y n，m (Ω) is a spherical harmonic function, n is the order, and m = -n, ..., n.

[0121] After simplification, the continuity conditions of the quasi-diffusion approximation equations can be obtained, namely the flow continuity boundary conditions and the flux continuity conditions:

[0122] ∫ Ω n·ΩΦ g (r, Ω)Y 0，0 (Ω)=n·J g (5)

[0123] Where, Φ g (r, Ω) represents the neutron angular flux density of the g energy group, J g This represents the neutron flux density of the g energy group.

[0124] ∫ Ω n·ΩΨ g (r, Ω)Y 1，0 (Ω)=n·E x，g (r)Φ g (r)

[0125] ∫ Ω n·ΩΨ g (r, Ω)Y 1，-1 (Ω)=n·E y，g (r)Φ g (r)

[0126] ∫ Ω n·ΩΨ g (r, Ω)Y 1，1 (Ω)=n·E z，g (r)Φ g (r) (6)

[0127] Since the quasi-diffusion equation and the traditional diffusion equation are similar in form, the main difference lies in the leakage terms. Traditional numerical methods based on diffusion theory can be improved and extended to the numerical solution of the three-dimensional quasi-diffusion equation, such as the analytical nodal method, nodal expansion method, variational nodal method, and Green's function method. The variational nodal method is one of the most successful nodal methods in nuclear reactor core neutronics calculations and is widely used in solving traditional diffusion equations. Considering future geometric applicability, this invention proposes to use the variational nodal method to achieve the numerical solution of the three-dimensional quasi-diffusion equation.

[0128] 2) Solving the quasi-diffusion equations in hexagonal geometry using the variational nodal block method

[0129] According to Galerkin's variational principle, a global functional F can be established for the Quasi-diffusion equation over the entire solution domain and its boundaries, which consists of several nodal blocks:

[0130]

[0131] The contribution of node v is:

[0132]

[0133] in, χ γg The net neutron flux density at the nodal boundary, dV represents the spatial integral over nodal v; γ is the boundary surface, φ g S is the neutron standard flux density. g For g-group neutron source terms, It is the Laplace operator.

[0134] For the neutron standard flux density φ in equation (8) g , Neutron source term S g and the net neutron flux density χ at the nodal boundary g The expansions are performed using the orthogonal basis functions within the hexagonal nodules, with the energy group number g omitted for simplicity:

[0135]

[0136]

[0137]

[0138] Among them, f i (r), h kγ (r) is the expansion polynomial of the space basis functions, and the space basis functions f i(r), h kγ (r) is a complete orthogonal polynomial:

[0139] ∫ v f i (r)f i (r)dv=δ ij

[0140] ∫ γ h iγ (r)h jγ (r)dГ=δ ij (10)

[0141] Where, δ ij For Kroneck's symbol.

[0142] Substituting equation (9) into equation (8) yields the relationship between the expansion coefficients of the block functional and the neutron flux density, neutron source, and net neutron flux density.

[0143] F v [φ, χ] = φ T Aφ-2φ T s+2φ T Mχ (11)

[0144] Where φ, s, and χ are vectors composed of the expansion coefficients of the neutron flux density, neutron source, and net neutron flux density; the formulas for calculating matrices A and M are:

[0145] A ij =P ij +δ ij V v ∑ r

[0146]

[0147] M ikγ =∫ γ f i (r)h iγ (r)dΓ

[0148] M = [M1, M2, M3...] (12)

[0149] After processing, the block-based neutron flow J can be obtained. + and J - The response relationship between the block-average neutron flux density φ.

[0150] J + =Bs+RJ -

[0151] φ=Hs-C(J + -J -(13)

[0152] Among them, B, R, H, and C are response matrices jointly determined by geometry and material properties.

[0153] Equation (13) indicates that the hexagonal segments are coupled through surface neutron flow. The continuity conditions in equations (5) and (6) should be fully considered when dealing with the quasi-diffusion equation. In cases such as... Figure 4 In the hexagonal block shown, for surfaces 1 and 2 perpendicular to the coordinate axis, the net neutron flux density in the normal direction is expressed by equation (14). For surfaces 3, 4, 5, and 6 not perpendicular to the coordinate axis, the net neutron flux density in the normal direction is expressed by equation (15). The corresponding flux continuity conditions are shown in (16) and (17).

[0154]

[0155]

[0156]

[0157] Where E xx E yy The Eddington factors in the x and y directions are respectively constant within the homogenized nodal, ∑ tr The sum of the absorption and scattering cross sections, Φ, represents the neutron flux density. The superscripts 1 and 2 indicate the numbers of two adjacent segments, with segment 1 located at the negative end of the normal direction and segment 2 at the positive end. Let be the surface neutron flux, and + and - represent the surface in the positive and negative directions of the normal, respectively.

[0158] Using equations (16) and (17) and the relationship between the neutron current and the net neutron current, it is easy to obtain the neutron current at the interface between adjacent nodes. Corrected expression:

[0159]

[0160] in,

[0161]

[0162] f u =E xx for face 1, 2

[0163] f u =(E xx cos 2 α+E yy cos 2 β) for face 3, 4, 5, 6

[0164] All the nodes are coupled through equation (13), and then the entire system can be solved based on the traditional source iteration method.

[0165] 3) Eddington factor calculation

[0166] This invention treats the Eddington factor as a minority homogenization parameter similar to other participants such as fission cross section and scattering cross section, and adopts the homogenization cross section calculation idea in the "two-step method" of core calculation. It uses the component calculation to obtain an angular flux density distribution, and then calculates the Eddington factor of the homogenization region.

[0167]

[0168] Where V is the volume of the homogenized region, and Ω is the angle; Ω u Ω v Let θ be the angle in each direction, ψ(r, E, Ω) be the neutron angular flux density, and Φ(r, E) be the neutron flux density.

[0169] For advanced reactor types with complex neutron energy spectra and strong neutron anisotropy, some assumptions in traditional thermal reactor component calculation programs are no longer applicable. Because the Monte Carlo method uses continuous energy point cross sections, it can handle complex nuclide resonance problems and has strong geometric description capabilities, making it a popular choice for generating homogenization cross section parameters in fast reactors in recent years. While Monte Carlo programs such as OpenMC and Serpent have built-in functions for calculating minority group homogenization cross sections, they lack the ability to calculate the Eddington factor. This study aims to investigate the Monte Carlo method for calculating the Eddington factor based on equation (19), modifying the Monte Carlo program source code to enable it to calculate the Eddington factor.

[0170] 4) Analysis of the computational accuracy and efficiency of this invention

[0171] To verify the computational accuracy and efficiency of quasi-diffusion theory under hexagonal component cores, benchmark problems of advanced reactors such as RBWR and fast reactors were used. These benchmark problems were analyzed computationally, and the quasi-diffusion method was compared with traditional diffusion and transport methods in terms of computational accuracy and efficiency through keff and power density distribution error calculations.

[0172] For novel reactors with hexagonal components exhibiting complex neutron spectra and strong neutron anisotropy, this invention establishes a high-precision, computationally efficient core neutronics calculation method based on the variational segmentation block method to solve quasi-diffusion under hexagonal geometry. This is an innovation in core neutronics calculation methods, overcoming the accuracy problems of diffusion equations and the efficiency problems of transport equations, and has significant implications and practical engineering value.

[0173] 2. Regarding the technical solution in 1, are there any other alternative solutions that can also achieve the purpose of the invention?

[0174] Other methods can be used to solve for the Eddington factor. For example, the component program based on the deterministic MOC method can also be modified to calculate the Eddington factor, and then combined with some minority group homogenization parameters to achieve the purpose of solving the quasi-diffusion equation.

[0175] 3. The key technical point of this invention is to calculate the Eddington factor using the Monte Carlo method and to numerically solve the quasi-diffusion equation under hexagonal geometry. The point to be protected by this invention is the use of the variational nodal block method to solve the quasi-diffusion equation under hexagonal geometry. Furthermore, the derivation and calculation of the Eddington factor are also points to be protected by this invention.

[0176] 4. The inventive aspects of this invention compared to the prior art.

[0177] First, the method for calculating the Eddington factor is innovative. Second, currently, the use of quasi-diffusion equations to solve core problems both domestically and internationally mainly targets 1-dimensional or 2-dimensional problems. Therefore, using the variational segmentation block method to solve the 3-dimensional quasi-diffusion equations under hexagonal components is also one of the innovative points of this invention.

[0178] 5. What principles and methods were used to achieve this?

[0179] (1) In view of the Eddington factor, a key parameter in the solution process of the quasi-diffusion equation and considering the complexity of the neutron energy spectrum in the reactor core, a "two-step method" is adopted. The source code of the Monte Carlo program Serpent is modified to enable it to calculate the Eddington factor.

[0180] (2) When using the Eddington factor expression, the first-order angular flux approximation is introduced, and finally the three-dimensional quasi-diffusion equation is established.

[0181] (3) Drawing on the current mature numerical solution algorithm for traditional diffusion equations—the variational block method—we study its method for handling the leakage term of the quasi-diffusion equation and establish a numerical solution model for the three-dimensional quasi-diffusion equation.

[0182] 6. First, when solving reactor core problems using existing technologies, traditional diffusion theory is usually still used. However, for new reactor types with characteristics such as complex energy spectrum, complex geometry, and strong neutron flux anisotropy, the calculation accuracy of diffusion theory is greatly biased. On the other hand, it is difficult to guarantee the calculation efficiency of using transport theory to solve the problem with high accuracy. Therefore, it is of great significance for the development and design of reactors to use the Eddington factor to solve the quasi-diffusion theory while considering both efficiency and accuracy.

[0183] 7. Technical solution.

[0184] (1) Calculation of Eddington factor

[0185] The Eddington factor is defined as follows:

[0186]

[0187] To complete the calculation of the Eddington factor, the flight direction Ω of each particle in the source code needs to be determined. u Ω v Ω w And the location of the neutron angle flux variable along this direction, by modifying the Serpent program source code to count the following variables in 4-space:

[0188]

[0189] The neutron standard flux φ can be calculated directly from the original program, so the Eddington factor is ultimately calculated using the following formula:

[0190]

[0191] (2) Deriving the quasi-diffusion equation based on the Eddington factor

[0192] Similar to the derivation of the P1 equation, integrating the time-independent neutron transport equation over Ω∈[0, 4π] and multiplying it by Ω and integrating over the entire angular space yields the neutron scalar flux equation and the neutron flux equation, respectively. It is worth noting that, unlike the derivation of the P1 equation, this derivation of the neutron flux equation does not introduce the assumption of a first-order approximation of the neutron angular flux density over the angular variable, and defines the Eddington factor as follows:

[0193]

[0194]

[0195] Where Ω represents the angle; Ω u Ω vLet ψ(r, E, Ω) represent the angles in each direction, φ(r, E) represent the neutron angular flux density, and Φ(r, E) represent the neutron flux density. u (r, E) represents the energy component in the u direction. These represent the unit vectors in each direction.

[0196] The expression for neutron flow is as follows:

[0197]

[0198] Where ∑ tr (r, E) is the neutron transport cross section, and J(r, E) is the neutron flux density.

[0199] Finally, the quasi-diffusion equation is obtained:

[0200]

[0201] Where, k eff χ(E) is the effective multiplication factor, v∑ is the fission energy spectrum. f (r, E′) is the neutron production cross section, ∑ s0 (r, E′→E) represents the scattering cross section from energy group E′ to energy group E. ∑ t (r, E) is the sum of the absorption cross section and the scattering cross section, used to calculate the consumption.

[0202] (3) Solving the quasi-diffusion equation based on the variational block method

[0203] According to Galerkin's variational principle, a global functional F can be established for the quasi-diffusion equation over the entire solution domain and its boundaries, which consists of several nodal blocks:

[0204]

[0205] The contribution of node v is:

[0206]

[0207] in, χ γg This represents the net neutron flux density at the nodal boundary.

[0208] For the neutron standard flux density φ in equation (27) g , Neutron source term S g and the net neutron flux density χ at the nodal boundary g The expansions are performed using the orthogonal basis functions within the hexagonal nodules, with the energy group number g omitted for simplicity:

[0209]

[0210]

[0211]

[0212] The spatial basis functions are complete orthogonal polynomials:

[0213] ∫ v f i (r)f i (r)dv=δ ij

[0214] ∫ γ h iγ (r)h jγ (r)dГ=δ ij (29)

[0215] Where, δ ij For Kroneck's symbol.

[0216] Substituting equation (28) into equation (27) yields the relationship between the expansion coefficients of the nodal functional and the neutron flux density, neutron source, and net neutron flux density.

[0217] F v [φ, χ] = φ T Aφ-2φ T s+2φ T Mχ (30)

[0218] Where φ, s, and χ are vectors composed of the expansion coefficients of the neutron flux density, neutron source, and net neutron flux density; the formulas for calculating matrices A and M are:

[0219] A ij =P ij +δ ij V v ∑ r

[0220]

[0221]

[0222] M = [M1, M2, M3…] (31)

[0223] After processing, the block-based neutron flow J can be obtained. + and J - The response relationship between the block-average neutron flux density φ moment.

[0224] J + =Bs+RJ -

[0225] φ=Hs-C(J + -J - (32)

[0226] Among them, B, R, H, and C are response matrices jointly determined by geometry and material properties.

[0227] Equation (32) indicates that the hexagonal segments are coupled through surface neutron flow. The continuity conditions in equations (5) and (6) should be fully considered when dealing with the quasi-diffusion equation. In cases such as... Figure 4 In the hexagonal block shown, for surfaces 1 and 2 perpendicular to the coordinate axis, the net neutron flux density in the normal direction is expressed by equation (33). For surfaces 3, 4, 5, and 6 not perpendicular to the coordinate axis, the net neutron flux density in the normal direction is expressed by equation (34). The corresponding flux continuity conditions are shown in (35) and (36).

[0228]

[0229]

[0230]

[0231] Where E xx E yy Within the homogenized blocks, the values ​​are constants. The superscripts 1 and 2 represent the numbers of two adjacent blocks, with block 1 located at the negative end of the normal direction and block 2 at the positive end. Let be the surface neutron flux, and + and - represent the surface in the positive and negative directions of the normal, respectively.

[0232] Using equations (35) and (36) and the relationship between the split neutron flow and the net neutron flow, it is easy to obtain the corrected expression for the split neutron flow at the interface between adjacent nodes:

[0233]

[0234] in,

[0235]

[0236] f u =E xx for face 1, 2

[0237] f u =(E xx cos 2 α+E yy cos 2 β) for face 3, 4, 5, 6

[0238] All the nodes are coupled through equation (32), and then the entire system can be solved based on the traditional source iteration method.

[0239] 8. This invention, through the derivation, calculation, and verification of the quasi-diffusion equation based on the variational nodal block method for solving hexagonal geometry, establishes a practical, efficient, and accurate method for calculating neutron flux in reactor cores. This method can be used to solve complex or simple reactor configurations with various hexagonal geometries, effectively overcoming the accuracy problems of traditional diffusion equation calculations and the efficiency problems of transport equations. It has significant implications and practical engineering value.

[0240] This invention proposes to solve the Quasi-diffusion equation based on a two-step core computation method. The conventional minority homogenization section and the Eddington factor, a key parameter in the Quasi-diffusion equation, are calculated using the Monte Carlo program SERPENT-2.1.26. Since the SERPENT program itself has a conventional minority homogenization section such as ∑... a ,∑ s ,∑ f It has the function to calculate parameters such as Eddington factor, but not Eddington factor. This invention formulates the following through the above method: Figure 6 The SERPENT program modification scheme is shown.

[0241] First, read the SERPENT source code and refer to the existing calculation process for the conventional homogenization cross-section variables. Taking the absorbing cross-section as an example, the variable in the code is defined as GCU_INF_ABS, which first needs to be defined in the header file location.h. Similarly, the variables related to the Eddington factor also need to be defined here, such as... Figure 7 As shown, GCU_INF_FLX_XX represents GCU_INF_FLX_XY indicates And so on.

[0242] Similar to the calculation of the absorption cross section, the quantities in the Eddington factor equation are calculated in the scoregc.h file: wgt*flx*xy, where wgt is the weight, flx is the neutron angular flux, and xy is the product of the angle between the angular flux direction and the x-axis and the y-axis. The result of this calculation is the numerator of the Eddington factor equation, and the denominator is calculated as wgt*flx.

[0243] To obtain the Eddington factor under an infinite energy spectrum, first add the microscopic components calculated above and store them in the Eddington factor calculation components, such as... Figure 9 middle.

[0244] Finally, the corresponding Eddington factor is output in the *.m file, such as... Figure 10 As shown.

[0245] II. Application Examples. To demonstrate the inventiveness and technical value of the technical solution of this invention, this section provides application examples of the technical solution of the claims on specific products or related technologies.

[0246] To demonstrate the inventiveness and technical value of this invention, and considering that the correctness of the Eddington factor is a crucial basis for the Quasi-diffusion equation, the calculation of the Eddington factor is first verified through examples:

[0247] Example 1:

[0248] As defined by the Eddington cloud factor, for problems involving anisotropic and isotropic neutron fluxes, the Eddington factor for the diagonal term should be approximately 1 / 3. Based on the modified Serpent source code, which calculates the Eddington factor, a spherical source was established with vacuum boundary conditions, UO2 fuel, and the energy group divided into two groups with a boundary energy of 0.625 MeV. The number of particles introduced was 10,000, and the calculation was iterated for 300 generations. After discarding the first 100 generations, the Eddington factor calculation results based on the Serpent output card are as follows:

[0249] Group 1 Results:

[0250]

[0251] Group 2 results:

[0252]

[0253] As can be seen from the calculation results above, the Eddington factor for the diagonal term is close to 1 / 3, but there are still some errors. In order to reduce the statistical error that may be caused by too few particles, the number of particles is increased to 100,000. The Eddington factor calculation results are as follows:

[0254] Group 1 Results:

[0255]

[0256] Group 2 results:

[0257]

[0258] The number of particles was increased to 1,000,000. The Eddington factor was calculated as follows:

[0259] Group 1 Results:

[0260]

[0261] Group 2 Results

[0262]

[0263] The above calculation results show that for a 3×3 Eddington factor tensor array, the calculation result for the diagonal term is almost equal to 1 / 3. As the number of particles increases from 10,000 to 100,000, the Eddington factor for the diagonal term becomes closer to 1 / 3. However, when the number of particles is further increased to 1,000,000, the error change is no longer significant. This indicates that within a certain range of particle counts, increasing the number of particles will reduce the statistical error. However, once the number of particles exceeds a certain value, the statistical error still exists and is difficult to reduce. This is basically consistent with the theoretical derivation of the calculated value in this invention. Therefore, it can be considered that the calculation of the Eddington factor using the Serpent redevelopment is successful. This technology can provide a convenient option for subsequent users to calculate the Eddington factor and also provides some reference value for other methods of calculating the Eddington factor.

[0264] To demonstrate the inventiveness and technical value of this invention, the Quasi-diffusion core neutron calculation method based on the variable segmented block method for hexagonal component cores is a key innovation of this invention. Secondly, an example verification is provided for the hexagonal block Quasi-diffusion program VNMQD based on the above method:

[0265] Example 2:

[0266] The program currently does not consider discontinuity factors. This example uses the 1D-RBWR benchmark problem to verify and analyze the program, such as... Figure 12 As shown;

[0267] The model has a side spacing of 19.7668 cm, radial total internal reflection, and an axial height of 134.3 cm. It is divided into 34 axial layers from bottom to top, with heights of 5*5.6 cm, 8*2.4125 cm, 8*6.5 cm, 8*3.5 cm, and 5*1.4 cm. The calculation employs a two-step method. First, the Monte Carlo program Serpent generates 12 groups of minority-group homogenization cross sections and Eddington factors. Each mesh layer generates a set of homogenization parameters, resulting in 34 homogenization cross sections generated by Serpent. Calculations are then performed using both axial total internal reflection and vacuum boundary conditions. The results from the Serpent program are used as a reference solution for comparison, including the effective core multiplication factor and the average neutron flux density per nodal.

[0268] Table 1. One-dimensional RBWR problem 12: group energy group partitioning

[0269]

[0270] 1) Axial total internal reflection boundary conditions

[0271] The results of VNMQD calculations under the total reflection boundary condition for the one-dimensional RBWR problem are shown in Table 2 below. The "no edd" case indicates that the Eddington factor is set to 1 / 3 during the calculation, i.e., it is converted to standard neutron diffusion calculation. Other minority group cross sections are consistent with VNMQD(edd). The comparison results show that, compared to standard diffusion calculations, VNMQD(edd) has a smaller error of 50.3 pcm in the core keff and the Serpent non-uniform solution, while VNMQD(no edd) has an error of -815.2 pcm. The calculation time for VNMQD(no edd) and VNMQD(edd) is approximately 8.2 s, which is much shorter than the 3029 s time for parallel calculation with 24 cores in Serpent.

[0272] Table 2. Comparison of keff results of VNMQD calculations under total internal reflection boundary conditions for the one-dimensional RBWR problem.

[0273]

[0274] This example further compares the neutron flux density of each block, using the Serpent statistics as a reference solution. To save space, this invention only lists the calculation results of different methods for groups 1, 4, 8, and 12, along with their comparison errors with the Serpent reference solution. The results are as follows: Figure 13 , Figure 14 , Figure 15 , Figure 16 As can be seen from the figure, the VNMQD(edd) calculation results agree better with the Serpent reference values. The maximum relative error of neutron flux in groups 1, 4, and 8 does not exceed 5%, while the maximum error of VNMQD(edd) results can reach 30%. For the neutron flux density of group 11, the calculation error of VNMQD(edd) is slightly larger, reaching 10%, but it is still much smaller than the calculation error of VNMQD(no edd) of 40%.

[0275] 2) Axial vacuum boundary conditions

[0276] For the same 1D problem, vacuum boundary conditions are applied both vertically and horizontally. Similar to the total internal reflection boundary condition, the Serpent Monte Carlo program calculates 34 homogenization cross sections for 12 energy groups under the vertical vacuum boundary conditions, and the Serpent calculation results are still selected as the reference solution. The core keff calculation results are shown in Table 3. Similar to the total internal reflection boundary condition, compared with the standard diffusion calculation VNMQD (no edd), the core keff calculated by VNMQD (edd) has a smaller error of -73 pcm compared to the Serpent non-homogeneous solution, while the error of VNMQD (no edd) reaches -1738 pcm. The calculation time of VNMQD (no edd) and VNMQD (edd) is about 6.2 s, which is much smaller than the 998 s time of the 24-core parallel calculation by Serpent.

[0277] Table 3. Comparison of keff results of VNMQD calculations under vacuum boundary conditions for the one-dimensional RBWR problem.

[0278]

[0279] To further verify the computational accuracy of the program, the neutron flux density distributions and errors of the nodal blocks in groups 1, 4, 8, and 12 were also compared. The results are as follows: Figure 17 , Figure 18 , Figure 19 , Figure 20 As shown in the figure, the VNMQD(edd) calculation results agree better with the serpent reference values. The maximum relative error of the neutron flux in groups 1, 4, and 8 does not exceed 10%, while the maximum error of the VNMQD(edd) results can reach 160%.

[0280] Example 3: To further verify the application of the VNMQD program in a hexagonal reactor core, this section is based on the BN-600 core distribution and fuel type (e.g., Figure 21 As shown, a supercomponent model with 7 components was established, such as... Figure 22 As shown, the supermodule model consists of two concentric rings of modules: the central module is the SHR control rod module, surrounded by LEZ fuel modules. The axial arrangement of the modules is consistent with the BN-600 benchmark. In the calculations, the axial boundary condition is set to vacuum, and the radial boundary condition is set to total internal reflection.

[0281] The calculation process adopts a two-step method for core calculation: 1) The Monte Carlo program SERPENT is used to perform component homogenization calculation to obtain the homogenized minority cross section and Eddington factor of the material; 2) Based on the obtained minority homogenized cross section, the VNMQD program is used to perform core calculation.

[0282] Because the boundaries of problems computed by the SERPENT program can only be defined as conventional shape boundaries (rectangles, hexagons, circles, and other non-complex shapes, such as...), Figure 17 (The boundary shown cannot be accurately constructed). Therefore, in actual calculations, this invention employs the following... Figure 23 The computational region shown is used for SERPENT component minority group section homogenization calculation. The structure of the 11 minority groups is shown in Table 4.

[0283] Table 4. Group structure of 11 minority groups

[0284]

[0285] Based on the 12-layer material region division along the component's axis, the axial heights from bottom to top are 30cm, 4.5cm, 15cm, 4.5cm, 23cm, 5.3cm, 41.15cm, 5.1cm, 41.15cm, 5.5cm, 29.7cm, and 30cm, respectively. Using SREPENT, minority-group homogenized cross sections of 24 materials were generated (i.e., 12 materials along the LEZ axis and 12 materials along the SHR axis).

[0286] Due to SERPENT's computational area ( Figure 23 The red area shown is inconsistent with the region calculated by the variational nodal method. Therefore, when comparing the calculation results, the results calculated by SERPENT are not used as the benchmark solution. Instead, the calculation results of the PN transport procedure based on the variational nodal method are used as the reference solution. This report only verifies the application of Quasi-diffusion theory to the hexagonal nodal method, and using the PN transport solution as the benchmark solution is reasonable. In diffusion and Quasi-diffusion calculations, the transport-corrected total cross section is used to calculate the diffusion coefficient, and a 0th-order scattering matrix is ​​used. In PN transport calculations, the total cross section and 0th and 1st-order scattering matrices are used. The calculation results of the effective multiplication coefficient of the 3D hypercomponent model are shown in Table 5.

[0287] Table 5. Keff calculation results for the 3D hypercomponent model

[0288]

[0289] As can be seen from the table, the keff values ​​for P5 and P7 are very close, indicating that the keff obtained by using the 5th-order spherical harmonic function expansion of the angle in this calculation problem is accurate enough and can be used as a benchmark solution. The comparison in the table shows that the keff error of the traditional diffusion calculation is -471 pcm, while the keff error of VNMQD is -259 pcm. The core calculation accuracy based on Quasi-diffusion is better than that of traditional diffusion, while the computational efficiency of the two is similar.

[0290] The average neutron flux densities of groups 1, 4, 8, and 11 in different axial sections of the SHR and LHR modules are compared below. Section numbers are: SHR from bottom to top (1-12), and LEZ from bottom to top (13-24). The results are as follows: Figure 24 , Figure 25 , Figure 26 , Figure 27 As shown in the comparison, the VNMQD(edd) calculation results are in better agreement with the reference solution. The neutron flux error for the vast majority of nodes is smaller than that calculated by VNMQD(no edd), and for some nodes, the error is equal to that of VNMQD(no edd). For this problem, due to the relatively weak neutron angular anisotropy, there is no significant difference between the VNMQD(no edd) and VNMQD(edd) calculation results, including the core keff.

[0291] Example 4:

[0292] To further verify the program, this section designs a core with 12 fuel rings based on the BN600 core, such as... Figure 28 As shown. The reactor core consists of 12 rings of fuel, with a radial vacuum boundary condition and an axial height of [missing information]. Figure 21 The division is the same. The core Keff calculation results are shown in Table 6. Since the Serpent model can accurately simulate this, the Serpent calculation results are selected as the reference solution. It can be seen that the conclusions drawn from the core calculation results are similar to those of the 3D meta-component model. Compared with the reference solution, the VNMQD(edd) calculation result has a smaller error than the VNMQD(no edd) result, which is only 11 pcm, while the VNMQD(no edd) result can reach -253 pcm. The computational efficiency of both is close to about 3 seconds, which is much smaller than the Monte Carlo calculation time.

[0293] Similarly, the average neutron flux densities of groups 1, 4, 7, and 11 calculated using different methods were compared. The group numbers are as follows: Figure 24 As shown, the calculation result of VNMQD(edd) has a smaller error than that of VNMQD(no edd), and the advantage of VNMQD(edd) is more obvious near the boundary.

[0294] Table 6. Keff calculation results of different programs under radial vacuum of 12 fuel core rings.

[0295]

[0296] III. Evidence of the Relevant Effects of the Embodiments. The embodiments of the present invention have achieved some positive effects during research and development or use, and indeed possess significant advantages compared to existing technologies. The following description, in conjunction with data, charts, and other materials from the experimental process, illustrates these advantages.

[0297] First, regarding Example 1, the experimental process mainly verifies the correctness of the Eddington factor calculation. As can be seen from the spherical source calculation results in this example, although the influence of statistical error is difficult to completely eliminate, the numerical results of the Eddington factor tensor matrix are basically in agreement with the theoretical value. This confirms the feasibility of using the Monte Carlo program Serpent to treat the Eddington factor as a minority group parameter, and provides some reference value for future calculations of the Eddington factor.

[0298] Secondly, for the axial total reflection boundary condition, compared to the standard diffusion calculation VNMQD (no edd), the error between the core Keff calculated by VNMQD (edd) and the Serpent non-uniform solution is smaller at 50.3 pcm, while the error of VNMQD (no edd) reaches -815.2 pcm. Furthermore, the calculation time for VNMQD (no edd) and VNMQD (edd) is roughly equivalent at approximately 8.2 s, far less than the 3029 s time for parallel calculation of 24 Serpent cores. Moreover, considering the comparison of neutron flux density, combined with... Figure 8 , Figure 9 , Figure 10 , Figure 11 It can be seen that the VNMQD(edd) calculation results agree better with the Serpent reference values. The maximum relative error of neutron flux in groups 1, 4, and 8 does not exceed 5%, while the maximum error of VNMQD(edd) results can reach 30%. For the neutron flux density of group 11, the calculation error of VNMQD(edd) is slightly larger, reaching 10%, but it is still much smaller than the calculation error of VNMQD(no edd) at 40%. For the axial vacuum boundary condition, similar to the total reflection boundary condition, compared with the standard diffusion calculation VNMQD(no edd), the error of the core keff calculated by VNMQD(edd) with the Serpent non-uniform solution is smaller at -73 pcm, while the error of VNMQD(no edd) reaches -1738 pcm. The calculation time of VNMQD(no edd) and VNMQD(edd) is about 6.2 s, which is much smaller than the 998 s of the parallel calculation time of 24 cores of Serpent. Comparing the neutron flux density distributions and errors of nodal blocks in groups 1, 4, 8, and 12, from... Figure 12 , Figure 13 , Figure 14As shown in Figure 15, the VNMQD(edd) calculation results agree better with the Serpent reference values. The maximum relative error of the neutron flux in groups 1, 4, and 8 does not exceed 10%, while the maximum error of the VNMQD(edd) results can reach 160%. In summary, for this benchmark case, regardless of whether the boundary conditions are axial total reflection or vacuum boundary conditions, the Quasi-diffusion program VNMQD, with the introduction of the Eddington factor, significantly outperforms the traditional diffusion equations in terms of accuracy and efficiency. This, to some extent, addresses the shortcomings of commonly used traditional diffusion methods for complex core calculations.

[0299] Secondly, regarding Example 3, a comparison in Table 5 shows that the keff error of traditional diffusion calculation is -471 pcm, while the keff error of VNMQD is -259 pcm. The core calculation accuracy based on Quasi-diffusion is superior to that of traditional diffusion, while their computational efficiency is similar. Furthermore, comparing the average neutron flux density of groups 1, 4, 8, and 11 in different axial sections of the SHR and LHR modules shows that the VNMQD(edd) calculation results are more consistent with the reference solution. The neutron flux error in most sections is smaller than that in the VNMQD(no edd) calculation results, and the error in some sections is equal to that in the VNMQD(no edd) calculation. For this problem, since the neutron angular anisotropy is not very strong, there is no significant difference between the VNMQD(no edd) and VNMQD(edd) calculation results. In summary, for this benchmark case, whether under axial total internal reflection boundary conditions or vacuum boundary conditions, the Quasi-diffusion program VNMQD significantly outperforms the traditional diffusion equations in terms of accuracy and efficiency when incorporating the Eddington factor. This, to some extent, addresses the shortcomings of commonly used traditional diffusion methods for complex core calculations.

[0300] Finally, for Example 4, it can be seen that the conclusions drawn from the core calculation results are similar to those in Examples 2 and 3. Compared with the reference solution, the VNMQD(edd) calculation result has a smaller error than VNMQD(no edd), only 11 pcm, while the error of VNMQD(no edd) can reach -253 pcm. The computational efficiency of both is approximately 3 seconds, far less than the Monte Carlo calculation time. Similarly, comparing the average neutron flux density of the components in groups 1, 4, 7, and 11 calculated by different methods, it can be seen that the VNMQD(edd) calculation result has a smaller error than VNMQD(no edd), and the advantage of VNMQD(edd) is more pronounced near the boundary. For this benchmark case, whether it is axial total reflection boundary condition or vacuum boundary condition, the Quasi-diffusion program VNMQD is significantly better than the traditional diffusion equation in terms of calculation accuracy and efficiency when introducing the Eddington factor for Quasi-diffusion. This to some extent solves the shortcomings of the commonly used traditional diffusion method for complex core calculations.

[0301] In summary, compared with existing technologies, this invention improves the Serpent program, enabling it to calculate the Eddington factor, and verifies the accuracy of its calculated values. This provides a certain reference value for future calculations of the Eddington factor. Furthermore, as shown in Cases 2, 3, and 4, the Quasi-diffusion core neutron calculation method based on the Eddington factor demonstrates that, compared with commonly used traditional diffusion equation solving and transport solving calculations which offer high accuracy but low efficiency, the accuracy of this Quasi-diffusion calculation program is significantly better than diffusion calculation, while its efficiency is similar to that of diffusion calculation. This has significant engineering practical value for calculations of novel and complex reactor cores.

[0302] It should be noted that embodiments of the present invention can be implemented in hardware, software, or a combination of both. The hardware portion can be implemented using dedicated logic; the software portion can be stored in memory and executed by a suitable instruction execution system, such as a microprocessor or dedicated-design hardware. Those skilled in the art will understand that the above-described devices and methods can be implemented using computer-executable instructions and / or included in processor control code, for example, such code provided on a carrier medium such as a disk, CD, or DVD-ROM, a programmable memory such as read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuitry such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field-programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of the above-described hardware circuitry and software, such as firmware.

[0303] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by any person skilled in the art within the scope of the technology disclosed in the present invention, within the spirit and principles of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for processing variable segment blocks for determining the core of complex reactor configurations, characterized in that, The variational block processing method for determining the core of complex reactor types includes the following steps: Step 1: Establish the Quasi-diffusion equations and corresponding boundary conditions under hexagonal geometry; Step 2: Transform the leakage term of the second-order tensor using the variable segmentation block method to establish a variable segmentation block numerical calculation method for Quasi-diffusion under hexagonal geometry; Step 3: The Monte Carlo program Serpent calculates the Eddington factor. Step four: Analyze the computational accuracy and efficiency of the quasi-diffusion equation; In step two, the variational nodal block method is used to transform the leakage term of the second-order tensor, establishing a variational nodal block numerical calculation method for Quasi-diffusion under hexagonal geometry, including: Based on Galerkin's variational principle, a global functional is established for the Quasi-diffusion equation over the entire solution domain, which consists of several nodal blocks, and its boundaries. F : (7) The contribution of node v is: (8) in, Net neutron flux density at the nodal boundary; Neutron standard flux density in equation (8) neutron source term Net neutron flux density at nodal boundaries The expansions are performed using the orthogonal basis functions within the hexagonal nodules, omitting the energy group number g: (9) The spatial basis functions are complete orthogonal polynomials: (10) in, The symbol for Kronecker; Substituting equation (9) into equation (8) yields the relationship between the expansion coefficients of the nodal functional and the neutron flux density, neutron source, and net neutron flux density: (11) in, It is a vector composed of the expansion coefficients of neutron flux density, neutron source, and net neutron flux density; a matrix. A and M The calculation formula is: (12) After processing, a block-based neutron stream is obtained. and Block average neutron flux density Response relationship between moments: (13) in, B , R , H , C The response matrix is ​​determined by both geometry and material properties. Equation (13) indicates that the hexagonal segments are coupled through surface neutron flux. When dealing with the quasi-diffusion equation, the flux continuity boundary conditions and flux continuity conditions should be fully considered. In the hexagonal segments, for surfaces 1 and 2 perpendicular to the coordinate axes, the net neutron flux density in the normal direction is expressed by Equation (14). For surfaces 3, 4, 5, and 6 not perpendicular to the coordinate axes, the net neutron flux density in the normal direction is shown by Equation (15). The corresponding flux continuity conditions are shown in (16) and (17). (14) (15) (16) (17) in, Within the homogenized blocks, the values ​​are constants. The superscripts 1 and 2 represent the numbers of two adjacent blocks, with block 1 located at the negative end of the normal direction and block 2 at the positive end. Let be the neutron flux of the surface, and + and - represent the positive and negative directions of the surface normal, respectively; Using equations (16) and (17) and the relationship between the split neutron flow and the net neutron flow, the corrected expression for the split neutron flow at the interface between adjacent nodes is obtained: (18) in, All the nodes are coupled through equation (13), and the entire system is solved based on the traditional source iteration method.

2. The variable segmentation block processing method for determining the core of complex reactor types as described in claim 1, characterized in that, The step one, establishing the Quasi-diffusion equations and corresponding boundary conditions under hexagonal geometry, includes: for time-independent continuous energy neutron transport equations... Integrals and multiplications Integrating over the entire angular space yields the neutron standard flux equation and the neutron flux equation, respectively. Unlike the derivation of the P1 equation, this derivation of the neutron flux equation does not introduce the assumption of a first-order approximation of the neutron angular flux density over the angular variable, and defines the Eddington factor as follows: (1) in, For angle; For angles in all directions, The neutron angular flux density, Neutron flux density This represents the energy component in the u direction. These represent the unit vectors in each direction; The expression for neutron flow is as follows: (2) in, This is the neutron transport cross section. Neutron flux density; Finally, the quasi-diffusion equation is obtained: (3) in, The effective value-added coefficient, For fission energy spectrum, To create a cross section for neutrons Indicates from energy group Scattered into the energy group scattering cross section; This is the sum of the absorption cross section and the scattering cross section, used for statistical analysis of consumption. The continuity condition in the quasi-diffusion approximation equation is based on angular flux. The function r is a continuous function at the interface between the two media, and can be approximated using the following set of conditions: (4) in, Continuous on the interface, Let n be a spherical harmonic function, and n be the order. ; After simplification, the flow continuity boundary conditions and flux continuity conditions of the quasi-diffusion approximation equation are obtained: (5) in, This represents the neutron angular flux density of the g-group. This represents the neutron flux density of the g-group. (6)。 3. The variable segmentation block processing method for determining the core of complex reactor types as described in claim 1, characterized in that, Step three, in which the Monte Carlo program Serpent calculates the Eddington factor, includes: Treating the Eddington factor as a small-group homogenization parameter similar to other parameters including fission cross section and scattering cross section, the Monte Carlo program SERPENT source code was modified to enable it to calculate the Eddington factor. First, the flight direction of each particle in the SERPENT source code was determined. And the location of the neutron angle flux variable along this direction, which can be statistically analyzed by modifying the program source code. The following variables within the space: ; And standard neutron flux The original program calculates the Eddington factor using the following formula: .

4. The variable segmentation block processing method for determining the core of complex reactor types as described in claim 1, characterized in that, The calculation accuracy and efficiency analysis in step four includes: to verify the calculation accuracy and efficiency of the quasi-diffusion theory under the hexagonal component core, advanced reactors, including RBWR and fast reactor benchmarks, are used to perform calculation analysis on the benchmarks. Through keff and power density distribution error calculation, a comparative analysis of the quasi-diffusion method with traditional diffusion and transport methods is conducted in terms of calculation accuracy and efficiency.

5. The variable segmentation block processing method for determining the core of complex reactor types as described in claim 1, characterized in that, The variable segmentation block processing method for determining the core of complex reactor types is based on the two-step core calculation method to solve the Quasi-diffusion equation. The conventional minority homogenization section and the Eddington factor, a key parameter in the Quasi-diffusion equation, are calculated using the Monte Carlo program SERPENT-2.1.

26. First, read the SERPENT source code and refer to the original conventional homogenization section variable calculation process in the code. The variable in the code is defined as GCU_INF_ABS, which first needs to be defined in the header file location.h. Similar to the calculation of the absorption cross section, the various quantities in the Eddington factor equation are calculated in the scoregc.h file: wgt*flx*xy, where wgt is the weight, flx is the neutron angular flux, and xy is the product of the angle between the angular flux direction and the x-axis and the y-axis. The result of this calculation is the numerator term in the Eddington factor equation, and the denominator term is calculated as wgt*flx. To obtain the Eddington factor under an infinite energy spectrum, first add the microscopic components calculated above and store them in the Eddington factor calculation components. Finally, the corresponding Eddington factor is output in the *.m file.

6. A variational nodal block processing system for equations under hexagonal geometry applying the variational nodal block processing method for determining the core of complex reactor types as described in any one of claims 1 to 5, characterized in that, The variational block processing system for equations under hexagonal geometry includes: The equation establishment and solution module is used to establish and solve three-dimensional quasi-diffusion equations. The Eddington factor calculation module is used to perform Eddington factor calculations. The computational accuracy and efficiency analysis module is used to analyze the computational accuracy and efficiency of the quasi-diffusion equation.

7. A computer device, characterized in that, The computer device includes a memory and a processor. The memory stores a computer program that, when executed by the processor, causes the processor to perform the steps of the variable segmentation block processing method for determining the core of a complex reactor as described in any one of claims 1 to 5.

8. A computer-readable storage medium storing a computer program that, when executed by a processor, causes the processor to perform the steps of the variable segment block processing method for determining the core of a complex reactor configuration as described in any one of claims 1 to 5.

9. A computer-readable storage medium storing instructions that, when executed on a computer, cause the computer to apply the variational block processing system for equations under hexagonal geometry as described in claim 6.

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