Continuous energy characteristic line neutron and photon transport calculation method applied to nuclear related device

Abstract

The invention discloses a continuous energy characteristic line neutron and photon transport calculation method applied to a nuclear-related device. The method comprises the following steps: 1, carrying out continuous energy deterministic theory neutron transport calculation, photon transport calculation and neutron and photon transport calculation on a nuclear-related facility; 2, generating characteristic line information on the basis of the geometric model of the nuclear facility, and performing source iteration flux initialization; 3, calculating the fission source of each energy section of the nuclear-related facility; 4, calculating a scattering source of each energy section of the nuclear-related facility; 5, calculating the average angular flux of the nuclear facility and the flux density of the neutron standard; 6, judging whether the flux of the ith energy section of the nuclear-related facility is converged or not, and if not, returning to the step 4; judging whether all energy segments are scanned or not, and if not, returning to the step 4; judging whether the flux of each energy section converges or not, and if not, returning to the step 4; and 7, calculating an effective value-added factor keff of the nuclear-related facility, judging whether the keff is converged or not, and if the keff is not converged, returning to the step 4. According to the method, neutron and photon transport calculation with a wide application range and high efficiency is realized.

Description

Technical Field

[0001] The present invention relates to the technical field of neutron transport process simulation calculation and reactor physics analysis, and in particular to a continuous energy characteristic line neutron and photon transport calculation method applied to nuclear-related devices. Background Art

[0002] The neutron and photon transport equations quantitatively describe the migration, absorption, scattering, and leakage of neutrons and photons in the medium, and are differential-integral equations of the neutron and photon flux density distribution functions. By quantitatively solving the neutron and photon transport equations, we can obtain the distribution functions of the neutron and photon flux density in the seven-dimensional phase space composed of time (1 degree of freedom), space (3 degrees of freedom), flight direction (2 degrees of freedom), and flight energy (1 degree of freedom). The corresponding technology is a key core technology in many scenarios such as nuclear reactor core physics calculations, nuclear radiation field assessment calculations, and radiation shielding protection design. It can be applied to neutron transport calculations, photon transport calculations, and neutron and photon transport calculations in all nuclear-related devices.

[0003] The existing neutron and photon transport calculation methods mainly include probabilistic methods and deterministic methods. The former is also called the Monte Carlo method, which mainly obtains the neutron and photon flux density distribution information by sampling statistics in the sample space. In essence, it is limited by the law of large numbers. Although the calculation accuracy can be high, the calculation efficiency is generally low. In contrast, the deterministic method mainly calculates by algebraic methods to establish discrete distribution functions of neutron and photon flux densities in discrete phase space, which is mainly limited by the grid division of phase space. Up to now, the deterministic method can achieve accurate and efficient calculations in the six dimensions of time, space, and flight direction. However, the energy of particle flight spans more than a dozen orders of magnitude (generally 10 in the scenarios of nuclear reactor core physics calculations and radiation protection calculations). -11 ~10 1 MeV), and in the middle super-thermal energy region, due to the resonance characteristics of the nuclide nuclear reaction cross section, both the excitation curve of the cross section and the energy spectrum of the neutron and photon flux density have dense resonance phenomena, which brings a subversive challenge to the phase space grid division of the deterministic method.

[0004] The existing deterministic methods mainly use the multi-group approximation. This approximation divides the neutron and photon energy range into G intervals, each of which is called an energy group; the cross-section data of continuous energy is read from the evaluation nuclear database, which needs to be processed by nuclear data first and converted into a multi-group cross-section database, and then the multi-group neutron and photon transport calculations are performed. In the process of making a multi-group cross-section database, it is necessary to first give the typical neutron and photon flux density energy spectrum in the target application scenario; then, in each energy group in the non-resonance energy region, the neutron and photon flux density energy spectrum is used as the weight function to average the microscopic cross-section of the interaction between neutrons and photon nuclei along the energy dimension to obtain the multi-group microscopic cross-section; for each energy group in the resonance energy region, the typical neutron and photon flux density energy spectrum is used to calculate the resonance parameters including the resonance integral and the resonance energy spectrum, and finally the multi-group microscopic cross-section and resonance parameters are stored in the multi-group database. When calculating actual problems, it is necessary to first adopt the resonance calculation method. Based on information such as the geometric structure and material composition of the actual problem, and using the resonance parameters in the multi-group database, the multi-group cross-section of the actual problem within the resonance energy group is calculated; then multi-group deterministic neutron and photon transport calculations are carried out to finally obtain the grouped neutron and photon flux density distribution.

[0005] The multi-group approximation brings a series of problems: (1) The production of the multi-group database depends on the typical neutron and photon flux density energy spectrum in the target application scenario. For different calculation problems, a set of applicable multi-group databases needs to be re-produced, which makes it impossible to analyze specific problems specifically, increasing the development and calculation costs of the reactor core. (2) The multi-group approximation method requires the calculation of the resonance self-screening effect in the resonance energy group to obtain the microscopic cross-section in the resonance energy group. The current resonance calculation mainly uses the equivalent theory for calculation, but the equivalent theory uses many assumptions and approximations. Each approximation and assumption will bring some usage restrictions, making the scope of use of this calculation method smaller and smaller, and its accuracy is inevitably limited. (3) If the calculation grid of the multi-group approximation is divided more finely to increase the scope of application, the resulting algebraic equation group will increase sharply, resulting in a significant decrease in calculation efficiency. Summary of the invention

[0006] In order to overcome the problems existing in the above-mentioned prior art, the purpose of the present invention is to provide a continuous energy characteristic line neutron and photon transport calculation method applied to nuclear-related devices, directly perform continuous energy deterministic solutions to neutron and photon transport equations under arbitrary geometry, completely eliminate the problems of multi-group database production and resonance calculation, and achieve high-scope applicability and high-efficiency neutron and photon transport calculations.

[0007] In order to achieve the above object, the present invention adopts the following technical solution:

[0008] The calculation method of neutron and photon transport of continuous energy characteristic lines applied to nuclear devices is as follows:

[0009] Step 1: Take the nuclear device as the calculation object, and perform continuous energy deterministic neutron transport calculation, photon transport calculation, and neutron and photon transport calculation on the nuclear device according to the requirements;

[0010] Step 2: Generate characteristic line information based on the geometric model of the nuclear device and perform source iteration flux initialization;

[0011] Step 3: Based on the basis function expansion method of continuous energy determinism, the neutron energy range that needs to be calculated for nuclear devices is divided into I neutron energy segments. The logarithmic energy range of the i-th neutron energy segment is [u i ,u i-1 ],u i ,i=0,1,…,I is the logarithmic energy of the neutron energy segment boundary point, and its continuous energy neutron and photon transport equation is written as:

[0012]

[0013] Where:

[0014] i—the i-th neutron energy band;

[0015] I—maximum neutron energy range number;

[0016] r—neutron position vector;

[0017] u i —neutron logarithmic energy in the i-th neutron energy band;

[0018] u i' —neutron logarithmic energy of the i'th neutron energy band;

[0019] u i-1 —neutron logarithmic energy in the i-1th neutron energy band;

[0020] u i'-1 —neutron logarithmic energy in the i'-1th neutron energy band;

[0021] s—scattering;

[0022] f—fission;

[0023] Ω—neutron flight direction vector;

[0024] Φ i (r,u,Ω)—the flux in the ith energy band with neutron position r, neutron energy u, and neutron flight direction Ω;

[0025] Σ t(r,u)—total cross section when the neutron position is r and the neutron energy is u;

[0026] v(u')—the average number of neutrons produced in each fission of a neutron with energy equal to u;

[0027] Σ s (r,u')—scattering cross section when the neutron position is r and the neutron energy is u';

[0028] f s (r,u'→u,Ω'→Ω)—the probability that the neutron position is r, the neutron energy is scattered from u' to u, and the neutron flight direction is scattered from Ω' to Ω;

[0029] Σ f (r,u'→u,Ω'→Ω)—the fission cross section when the neutron position is r, the neutron energy changes from u' to u, and the neutron flight direction changes from Ω' to Ω;

[0030] Φ i′ (r, u′, Ω′)—the flux in the ith energy segment with neutron position r, neutron energy u′, and neutron flight direction Ω′;

[0031] For the i-th energy segment, according to the basis function expansion method of continuous energy determinism, an orthogonal basis function system is selected: satisfy:

[0032]

[0033] Where:

[0034] n—the expansion order is n;

[0035] n'—the expansion order is n';

[0036] P n (u)—the nth-order orthogonal basis function when the neutron energy is u;

[0037] P n' (u)—orthogonal basis function of order n' when the neutron energy is u;

[0038] N i —The maximum expansion order of the i-th energy segment;

[0039] N i' —The maximum expansion order of the i'th energy segment;

[0040] Using this orthogonal basis function system, the neutron flux density function is expanded to obtain:

[0041]

[0042] Where:

[0043] Φ i,n (r,Ω)—the flux expansion moment with expansion order n, spatial position vector r, and neutron flight direction Ω in the i-th energy band;

[0044] Φ i',n (r,Ω')—flux expansion moment in the ith energy band with expansion order n, spatial position vector r, and neutron flight direction Ω';

[0045] Projecting formula (3) onto the coordinate axes of each basis function in the function space composed of orthogonal basis functions, we get:

[0046]

[0047] in:

[0048]

[0049] Where:

[0050] Σ t,inn' (r)—average total cross section in the n'th basis function space when the neutron position is r and the expansion order is n in the i-th energy segment;

[0051] Σ s,ii'nn' (r,Ω,Ω')—the average scattering cross section in the n'th basis function space when the neutron position is r and the neutron flight direction is Ω, the expansion order of scattering from the i-th energy segment to the i'th energy segment is n;

[0052] F f,ii'nn' (r,Ω,Ω')—When the neutron position is r and the neutron flight direction is Ω, the expansion order from the i-th energy segment to the i'th energy segment is n, and the average scattering cross section in the n'th basis function space;

[0053] S in' —The fixed source intensity in the n'th basis function space in the ith energy segment;

[0054] N i —The maximum expansion order of the i-th energy segment;

[0055] Write all equations for different n' into a system of equations, and write the system of equations in matrix form, and we get:

[0056]

[0057] in:

[0058]

[0059] Where:

[0060] Φ i (r,Ω)—the flux matrix in the i-th energy segment when the neutron position is r and the neutron flight direction is Ω;

[0061] Φ i' (r,Ω)—the flux matrix in the i'th energy segment when the neutron position is r and the neutron flight direction is Ω;

[0062] Σ t,i (r)—total cross section matrix in the ith energy band with neutron position r;

[0063] Σ s,ii' (r,Ω)—scattering cross section matrix of neutron at position r, scattered from the i-th energy band to the i'th energy band;

[0064] F f,ii' (r,Ω)—scattering cross section matrix of neutron position r, fission from the i-th energy segment to the i'th energy segment;

[0065] S i —The fixed source matrix in the i-th energy segment;

[0066] Project the three-dimensional space position vector r into the two-dimensional space position vector s, and we have r = ssin(θ e ), where s is the position vector in two-dimensional space, θ e is the polar angle of the e-th polar angle direction, considering the expression of the directional derivative:

[0067]

[0068] In the two-dimensional space region h, along the j-th characteristic line in the f-th angular direction, equation (6) is written as:

[0069] Where:

[0070] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial angle direction and the e-th polar angle direction, at s, the neutron flight direction is Ω, and the expansion order of the i-th energy segment is n i neutron flux;

[0071] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the expansion order is n i Total cross-section of

[0072] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the neutron flight direction is scattered from Ω to Ω' and the expansion order is n i The neutron scattering cross section of

[0073] —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth radial angle direction and the eth polar angle direction, at s, the neutron flight direction from Ω fission to Ω' has an expansion order of n i neutron fission cross section;

[0074] Formula (9) is combined into a matrix form, namely:

[0075]

[0076] The above results in the characteristic line equation of the jth characteristic line along the fth direction in the two-dimensional space region h of the nuclear device;

[0077] At this time, in the two-dimensional space region h of the nuclear device, the fission source of the jth characteristic line along the fth direction is expressed as:

[0078]

[0079] Calculate the fission sources in each energy range;

[0080] Step 4: The scattering source of the jth characteristic line along the fth direction in the two-dimensional space region h of the nuclear device is expressed as:

[0081]

[0082] The scattering source term is expressed as:

[0083]

[0084] Using the Legendre addition theorem and the orthogonality of spherical harmonics, the expression of the scattering source is obtained as follows:

[0085]

[0086] In formula (14), Σ s,m,i'→i,hefj (s) has been obtained in the continuous energy neutron transport calculation, and the scattering source in the i-th energy band is calculated using equation (14).

[0087] Step 5: Assume that the total cross-section of the nuclear device is constant in the two-dimensional space region h, and assume that the neutron source is also constant in the two-dimensional space region h, and the flux is isotropic, then we get:

[0088]

[0089] in:

[0090]

[0091] Q in',hefj (s,Ω')—the total neutron source term with the order n of the expansion of the i-th energy segment in the neutron flight direction Ω', located at s, in the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction in the two-dimensional space region h;

[0092] Formula (11) is combined into a matrix form and expressed as:

[0093]

[0094] make:

[0095]

[0096] Where:

[0097] Q i,hefj —The total neutron source term of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial direction in the two-dimensional space region h;

[0098] From formula (14), we can see that formula (12) can be written as:

[0099]

[0100] Where:

[0101] Φ i,hefj —The neutron flux density of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial angle direction in the two-dimensional space region h;

[0102] Σ t,i,hefj —The total cross section of the i-th energy segment in the j-th characteristic line along the polar angle of the e-th polar angle direction in the f-th radial direction in the two-dimensional space region h;

[0103] Calculate the matrix Σ through matrix similarity transformation t,i,hefj All eigenvalues ​​of and its eigenvector And construct the matrix:

[0104]

[0105] For the matrix Σ t,i,hefj Perform a similarity transformation:

[0106]

[0107] Where:

[0108] Λ t,i,hefj —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, the total cross-sectional matrix Σ t,i,hefj A diagonal matrix composed of the eigenvalues ​​of ;

[0109] B t,i,hefj —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth angular direction and the eth polar angle direction, N i The total cross-section matrix Σ consists of linearly independent eigenvectors t,i,hefj Reversible matrices for similarity transformations;

[0110] λ n —Total cross-section matrix Σ t,i,hefj The nth eigenvalue of ;

[0111] ν n —Total cross-section matrix Σ t,i,hefj The nth linearly independent eigenvector of ;

[0112] After sorting, we can get:

[0113]

[0114] Where:

[0115]

[0116] Y(s)—total cross-section matrix Σ t,i,hefj The product of the reversible matrix of similarity transformation and the flux moment vector;

[0117] y n (s)—the nth value of Y(s);

[0118] The general solution of equation (23) is obtained to obtain the outgoing neutron flux density:

[0119]

[0120] Integrate equation (25) along the characteristic line segment and arrange it to obtain the expression of average neutron angular flux density:

[0121]

[0122] The average flux of all characteristic line segments in the f direction in the two-dimensional space region h is summed according to the volume weight to obtain the average angular flux in the f direction in the two-dimensional space region h:

[0123]

[0124] Where:

[0125] sh,f,j —The length of the jth characteristic line along the fth angular direction in the two-dimensional space region h;

[0126] δA f — cross-sectional area of ​​the characteristic line in the f direction;

[0127] V h —the volume of a two-dimensional space region h;

[0128] —In the two-dimensional space region h, the nth order average angular flux on the ith energy segment of the jth characteristic line along the fth angular direction

[0129] In actual calculation, it is necessary to correct the length of the characteristic line segment, that is:

[0130]

[0131] After obtaining the average angular flux of the polar angle e and the argument angle f in the two-dimensional space region h of the nuclear device by formula (27), the neutron standard flux density in the two-dimensional space region h is obtained by summing the angle weights of the argument angle and the polar angle of the neutron angular flux density:

[0132]

[0133] ω e —Polar angle direction weight;

[0134] ω f —Arg direction weight.

[0135] A characteristic line scan is then performed to calculate the neutron flux density.

[0136] Step 6: Determine whether the i-th energy segment flux of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device is converged. If not, return to step 4;

[0137] Determine whether all energy segments of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device have been scanned, and if not, return to step 4;

[0138] Determine whether the fluxes of each energy segment of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device are converged. If not, return to step 4;

[0139] Step 7: Calculate the effective value-added factor k of the nuclear facility eff , and determine k eff Whether it converges. If not, return to step 4.

[0140] Preferably, the nuclear-related devices include reactor cores, radiation shielding layers, and medical radiation equipment.

[0141] Compared with the prior art, the present invention has the following advantages:

[0142] 1) Since the present invention develops a characteristic line equation calculation method based on the continuous energy deterministic neutron and photon transport equation, it improves the current technology of the continuous energy deterministic neutron and photon transport equation calculation method, so that the technology of the continuous energy deterministic neutron and photon transport equation calculation can be applied to the reactor core physics analysis program.

[0143] 2) Since the present invention develops a characteristic line equation calculation method based on the continuous energy deterministic neutron and photon transport equations, it fundamentally solves the drawbacks of the conventional characteristic line equation calculation method based on the multi-group approximation method, and can analyze specific problems specifically, so that the reactor core physics analysis program has a wider scope of application.

[0144] 3) Since the present invention develops a characteristic line equation calculation method based on the continuous energy deterministic neutron and photon transport equations, the present invention needs to calculate fewer equations than the reactor core physics analysis program developed based on the hyperfine group theory and Monte Carlo theory at the same accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0145] Figure 1 The figure is a flow chart of the method of the present invention. DETAILED DESCRIPTION

[0146] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0147] like Figure 1 As shown, the present invention is applied to the calculation method of neutron and photon transport of continuous energy characteristic lines of nuclear-related devices. The nuclear-related device of this embodiment is illustrated by taking a reactor core assembly as an example. The specific steps are as follows:

[0148] Step 1: Take the reactor core assembly as the calculation object, and perform continuous energy deterministic neutron transport calculation, photon transport calculation, and neutron and photon transport calculation on the reactor core assembly according to the requirements: According to the continuous energy evaluation nuclear database, use the function expansion technology to numerically discretize the continuous energy neutron and photon transport equations, and give the continuous energy neutron flux density and nuclear reaction rate distribution;

[0149] Step 2: Generate characteristic line information based on the geometric model of the reactor core assembly and perform source iteration flux initialization;

[0150] Step 3: According to the basis function expansion method of continuous energy determinism, the neutron energy range of the calculation object to be calculated is divided into I neutron energy segments. The logarithmic energy range of the i-th neutron energy segment is [u i,u i-1 ],u i ,i=0,1,…,I is the logarithmic energy of the neutron energy segment boundary point, and its continuous energy neutron and photon transport equation is written as:

[0151]

[0152] Where:

[0153] i—the i-th neutron energy band;

[0154] I—maximum neutron energy range number;

[0155] r—neutron position vector;

[0156] u i —neutron logarithmic energy in the i-th neutron energy band;

[0157] u i' —neutron logarithmic energy of the i'th neutron energy band;

[0158] u i-1 —neutron logarithmic energy in the i-1th neutron energy band;

[0159] u i'-1 —neutron logarithmic energy in the i'-1th neutron energy band;

[0160] s—scattering;

[0161] f—fission;

[0162] Ω—neutron flight direction vector;

[0163] Φ i (r,u,Ω)—the flux in the ith energy band with neutron position r, neutron energy u, and neutron flight direction Ω;

[0164] Σ t (r,u)—total cross section when the neutron position is r and the neutron energy is u;

[0165] v(u')—the average number of neutrons produced in each fission of a neutron with energy equal to u;

[0166] Σ s (r,u')—scattering cross section when the neutron position is r and the neutron energy is u';

[0167] f s (r,u'→u,Ω'→Ω)—the probability that the neutron position is r, the neutron energy is scattered from u' to u, and the neutron flight direction is scattered from Ω' to Ω;

[0168] Σ f(r,u'→u,Ω'→Ω)—the fission cross section when the neutron position is r, the neutron energy changes from u' to u, and the neutron flight direction changes from Ω' to Ω;

[0169] Φ i′ (r, u′, Ω′)—the flux in the ith energy segment with neutron position r, neutron energy u′, and neutron flight direction Ω′;

[0170] For the i-th energy segment, according to the basis function expansion method of continuous energy determinism, an orthogonal basis function system is selected: satisfy:

[0171]

[0172] Where:

[0173] n—the expansion order is n;

[0174] n'—the expansion order is n';

[0175] P n (u)—the nth-order orthogonal basis function when the neutron energy is u;

[0176] P n' (u)—orthogonal basis function of order n' when the neutron energy is u;

[0177] N i —The maximum expansion order of the i-th energy segment;

[0178] N i' —The maximum expansion order of the i'th energy segment;

[0179] Using this orthogonal basis function system, the neutron flux density function is expanded to obtain:

[0180]

[0181] Where:

[0182] Φ i,n (r,Ω)—the flux expansion moment with expansion order n, spatial position vector r, and neutron flight direction Ω in the i-th energy band;

[0183] Φ i',n (r,Ω')—the flux expansion moment with expansion order n, spatial position vector r and neutron flight direction Ω in the i-th energy band;

[0184] Project equation (3) onto the coordinate axes of each basis function in the function space composed of orthogonal basis functions, that is, multiply both sides by P n' (u), and then perform integral averaging in the logarithmic energy interval. For the n'th basis function, we get:

[0185]

[0186] in:

[0187]

[0188] Where:

[0189] Σ t,inn' (r)—average total cross section in the n'th basis function space when the neutron position is r and the expansion order is n in the i-th energy segment;

[0190] Σ s,ii'nn' (r,Ω,Ω')—the average scattering cross section in the n'th basis function space when the neutron position is r and the neutron flight direction is Ω, the expansion order of scattering from the i-th energy segment to the i'th energy segment is n;

[0191] F f,ii'nn' (r,Ω,Ω')—When the neutron position is r and the neutron flight direction is Ω, the expansion order from the i-th energy segment to the i'th energy segment is n, and the average scattering cross section in the n'th basis function space;

[0192] S in' —The fixed source intensity in the n'th basis function space in the ith energy segment;

[0193] N i —The maximum expansion order of the i-th energy segment;

[0194] Write all equations for different n' into a system of equations, and write the system of equations in matrix form, and we get:

[0195]

[0196] in:

[0197]

[0198] Where:

[0199] Φ i (r,Ω)—the flux matrix in the i-th energy segment when the neutron position is r and the neutron flight direction is Ω; Φ i' (r,Ω)—the flux matrix in the i'th energy segment when the neutron position is r and the neutron flight direction is Ω; Σ t,i (r)—total cross section matrix in the ith energy band with neutron position r;

[0200] Σ s,ii' (r,Ω)—scattering cross section matrix of neutron at position r, scattered from the i-th energy segment to the i'th energy segment; F f,ii'(r,Ω)—scattering cross section matrix of neutron position r, fission from the i-th energy segment to the i'th energy segment;

[0201] S i —The fixed source matrix in the i-th energy segment;

[0202] Equation (6) is the continuous energy neutron and photon transport equation after function expansion;

[0203] The calculation of the reactor core assembly requires solving the two-dimensional neutron and photon transport equations. Taking into account the polar angles of the neutron and photon flight directions, the three-dimensional position vector r is projected into a two-dimensional spatial position vector s, and r = ssin(θ e ), where s is the position vector in two-dimensional space, θ e is the polar angle of the e-th polar angle direction, considering the expression of the directional derivative:

[0204]

[0205] In the two-dimensional space region h, along the j-th characteristic line in the f-th angular direction, equation (6) is written as:

[0206] Where:

[0207] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial angle direction and the e-th polar angle direction, at s, the neutron flight direction is Ω, and the expansion order of the i-th energy segment is n i neutron flux;

[0208] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the expansion order is n i Total cross-section of

[0209] —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the neutron flight direction is scattered from Ω to Ω' and the expansion order is n i The neutron scattering cross section of

[0210] —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth radial angle direction and the eth polar angle direction, at s, the neutron flight direction from Ω fission to Ω' has an expansion order of n i neutron fission cross section;

[0211] Formula (9) is combined into a matrix form, namely:

[0212]

[0213] The above results in the characteristic line equation of the jth characteristic line along the fth direction in the two-dimensional space region h of the reactor core assembly;

[0214] At this time, in the two-dimensional space region h of the reactor core assembly, the fission source of the jth characteristic line along the fth direction is expressed as:

[0215]

[0216] Calculate the fission sources in each energy range;

[0217] Step 4: The scattering source of the jth characteristic line along the fth direction in the two-dimensional space region h of the reactor core assembly is expressed as:

[0218]

[0219] After expanding the scattering probability function and the neutron angular flux density, the final scattering source term is expressed as:

[0220] Using the Legendre addition theorem and the orthogonality of spherical harmonics, the expression of the scattering source is obtained as follows:

[0221]

[0222] In formula (14), Σ s,m,i'→i,hefj (s) has been obtained in the continuous energy neutron transport calculation, and the scattering source in the i-th energy band is calculated using equation (14).

[0223] Step 5: When the total neutron source term is known, assume that the total cross-section of the reactor core assembly is constant in the two-dimensional space region h, that is, it does not change with s. Assume that the neutron source is also constant in the two-dimensional space region h, that is, the flat source approximation, and the flux is isotropic, which gives:

[0224]

[0225] in:

[0226]

[0227] Q in',hefj (s,Ω')—the total neutron source term with the order n of the expansion of the i-th energy segment in the neutron flight direction Ω', located at s, in the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction in the two-dimensional space region h;

[0228] Formula (11) is combined into a matrix form and expressed as:

[0229]

[0230] make:

[0231]

[0232] Where:

[0233] Q i,hefj —The total neutron source term of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial direction in the two-dimensional space region h;

[0234] From formula (14), we can see that formula (12) can be written as:

[0235]

[0236] Where:

[0237] Φ i,hefj —The neutron flux density of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial angle direction in the two-dimensional space region h;

[0238] Σ t,i,hefj —The total cross section of the i-th energy segment in the j-th characteristic line along the polar angle of the e-th polar angle direction in the f-th radial direction in the two-dimensional space region h;

[0239] Since the matrix Σ i,t,hefj It can be calculated based on the energy interval and its basis function, and it is taken as a known parameter. Equation (15) is a first-order non-homogeneous differential equation system with known incident boundary conditions. Through the matrix Σ t,i,hefj Coupled together; so the matrix Σ is obtained by similarity transformation of the matrix t,i,hefj All eigenvalues ​​of and its eigenvector And construct the matrix:

[0240]

[0241] For the matrix Σ t,i,hefj Perform a similarity transformation:

[0242]

[0243] Where:

[0244] Λ t,i,hefj —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, the total cross-sectional matrix Σ t,i,hefj A diagonal matrix composed of the eigenvalues ​​of ;

[0245] B t,i,hefj —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth angular direction and the eth polar angle direction, N i The total cross-section matrix Σ consists of linearly independent eigenvectors t,i,hefj Reversible matrices for similarity transformations;

[0246] λ n —Total cross-section matrix Σ t,i,hefj The nth eigenvalue of ;

[0247] ν n —Total cross-section matrix Σ t,i,hefj The nth linearly independent eigenvector of ;

[0248] After the arrangement, the coefficient matrix in equation (15) has become a diagonal matrix, and the equations are no longer coupled with each other, that is:

[0249]

[0250] Where:

[0251]

[0252] Y(s)—total cross-section matrix Σ t,i,hefj The product of the reversible matrix of similarity transformation and the flux moment vector;

[0253] y n (s)—the nth value of Y(s);

[0254] The general solution of equation (23) is obtained to obtain the outgoing neutron flux density:

[0255]

[0256] Integrate equation (25) along the characteristic line segment and arrange it to obtain the expression of average neutron angular flux density:

[0257]

[0258] The average flux of all characteristic line segments in the f direction in the two-dimensional space region h of the reactor core assembly is summed according to the volume weight to obtain the average angular flux in the f direction in the two-dimensional space region h of the reactor core assembly:

[0259]

[0260] Where:

[0261] s h,f,j —The length of the jth characteristic line along the fth angular direction in the two-dimensional space region h;

[0262] δA f — cross-sectional area of ​​the characteristic line in the f direction;

[0263] V h —the volume of a two-dimensional space region h;

[0264] —In the two-dimensional space region h, the nth order average angular flux on the ith energy segment of the jth characteristic line along the fth angular direction

[0265] Several rectangles are used to approximate the volume of the real geometry, but the volume of the cuboids is not equal to the volume of the real geometry. Therefore, in the actual calculation, the length of the characteristic line segment needs to be corrected, that is:

[0266]

[0267] After the average angular flux of the polar angle e and the argument angle f in the two-dimensional space region h of the reactor core assembly is obtained by formula (27), the neutron standard flux density in the two-dimensional space region h is obtained by summing the angle weights of the argument angle and the polar angle of the neutron angular flux density:

[0268]

[0269] ω e —Polar angle direction weight;

[0270] ω f —Arg direction weight.

[0271] A characteristic line scan is then performed to calculate the neutron flux density of the reactor core components.

[0272] Step 6: Determine whether the i-th energy segment flux of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the reactor core assembly converges, if not, return to step 4; determine whether all energy segments of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the reactor core assembly have been scanned, if not, return to step 4;

[0273] Determine whether the fluxes of each energy segment of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the reactor core assembly converge, and if not, return to step 4;

[0274] Step 7: Calculate the effective multiplication factor k of the reactor core assembly eff , and determine k eff Whether it converges. If not, return to step 4.

Claims

1. A method for calculating the transport of neutrons and photons using continuous energy characteristic lines of nuclear-related devices, characterized by: Here are the steps: Step 1: Take the nuclear device as the calculation object, and perform continuous energy deterministic neutron transport calculation, photon transport calculation, and neutron and photon transport calculation on the nuclear device according to the requirements; Step 2: Generate characteristic line information based on the geometric model of the nuclear device and perform source iteration flux initialization; Step 3: Based on the basis function expansion method of continuous energy determinism, the neutron energy range that needs to be calculated for nuclear devices is divided into I neutron energy segments. The logarithmic energy range of the i-th neutron energy segment is [u i ,u i-1 ],u i ,i=0,1,…,I is the logarithmic energy of the neutron energy segment boundary point, and its continuous energy neutron and photon transport equation is written as: Where: i—the i-th neutron energy band; I—maximum neutron energy range number; r—neutron position vector; u i —neutron logarithmic energy in the i-th neutron energy band; u i' —neutron logarithmic energy of the i'th neutron energy band; u i-1 —neutron logarithmic energy in the i-1th neutron energy band; u i'-1 —neutron logarithmic energy in the i'-1th neutron energy band; s—scattering; f—fission; Ω—neutron flight direction vector; Φ i (r,u,Ω)—the flux in the ith energy band with neutron position r, neutron energy u, and neutron flight direction Ω; Σ t (r,u)—total cross section when the neutron position is r and the neutron energy is u; v(u')—the average number of neutrons produced in each fission of a neutron with energy equal to u; Σ s (r,u')—scattering cross section when the neutron position is r and the neutron energy is u'; f s (r,u'→u,Ω'→Ω)—the probability that the neutron position is r, the neutron energy is scattered from u' to u, and the neutron flight direction is scattered from Ω' to Ω; Σ f (r,u'→u,Ω'→Ω)—the fission cross section when the neutron position is r, the neutron energy changes from u' to u, and the neutron flight direction changes from Ω' to Ω; Φ i ′(r, u′, Ω′)—the flux in the ith energy segment with neutron position r, neutron energy u′, and neutron flight direction Ω′; For the i-th energy segment, according to the basis function expansion method of continuous energy determinism, an orthogonal basis function system is selected: satisfy: Where: n—the expansion order is n; n'—the expansion order is n'; P n (u)—the nth-order orthogonal basis function when the neutron energy is u; P n' (u)—orthogonal basis function of order n' when the neutron energy is u; N i —The maximum expansion order of the i-th energy segment; N i' —The maximum expansion order of the i'th energy segment; Using this orthogonal basis function system, the neutron flux density function is expanded to obtain: Where: Φ i,n (r,Ω)—the flux expansion moment with expansion order n, spatial position vector r, and neutron flight direction Ω in the i-th energy band; Φ i',n (r,Ω')—flux expansion moment in the ith energy band with expansion order n, spatial position vector r, and neutron flight direction Ω'; Projecting formula (3) onto the coordinate axes of each basis function in the function space composed of orthogonal basis functions, we get: in: Where: Σ t,inn' (r)—average total cross section in the n'th basis function space when the neutron position is r and the expansion order is n in the i-th energy segment; Σ s,ii'nn' (r,Ω,Ω')—the average scattering cross section in the n'th basis function space when the neutron position is r and the neutron flight direction is Ω, the expansion order of scattering from the i-th energy segment to the i'th energy segment is n; F f,ii'nn' (r,Ω,Ω')—When the neutron position is r and the neutron flight direction is Ω, the expansion order from the i-th energy segment to the i'th energy segment is n, and the average scattering cross section in the n'th basis function space; S in' —The fixed source intensity in the n'th basis function space in the ith energy segment; N i —The maximum expansion order of the i-th energy segment; Write all equations for different n' into a system of equations, and write the system of equations in matrix form, and we get: in: Where: Φ i (r,Ω)—the flux matrix in the i-th energy segment when the neutron position is r and the neutron flight direction is Ω; Φ i' (r,Ω)—the flux matrix in the i'th energy segment when the neutron position is r and the neutron flight direction is Ω; Σ t,i (r)—total cross section matrix in the ith energy band with neutron position r; Σ s,ii' (r,Ω)—scattering cross section matrix of neutron at position r, scattered from the i-th energy band to the i'th energy band; F f,ii' (r,Ω)—scattering cross section matrix of neutron position r, fission from the i-th energy segment to the i'th energy segment; S i —The fixed source matrix in the i-th energy segment; Project the three-dimensional space position vector r into the two-dimensional space position vector s, and we have r = ssin(θ e ), where s is the position vector in two-dimensional space, θ e is the polar angle of the e-th polar angle direction, considering the expression of the directional derivative: In the two-dimensional space region h, along the j-th characteristic line in the f-th angular direction, equation (6) is written as: Where: —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial angle direction and the e-th polar angle direction, at s, the neutron flight direction is Ω, and the expansion order of the i-th energy segment is n i neutron flux; —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the expansion order is n i Total cross-section of —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, in the i-th energy segment at s, the neutron flight direction is scattered from Ω to Ω' and the expansion order is n i The neutron scattering cross section of —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth radial angle direction and the eth polar angle direction, at s, the neutron flight direction from Ω fission to Ω' has an expansion order of n i neutron fission cross section; Formula (9) is combined into a matrix form, namely: The above results in the characteristic line equation of the jth characteristic line along the fth direction in the two-dimensional space region h of the nuclear device; At this time, in the two-dimensional space region h of the nuclear device, the fission source of the jth characteristic line along the fth direction is expressed as: Calculate the fission sources in each energy range; Step 4: The scattering source of the jth characteristic line along the fth direction in the two-dimensional space region h of the nuclear device is expressed as: The scattering source term is expressed as: Using the Legendre addition theorem and the orthogonality of spherical harmonics, the expression of the scattering source is obtained as follows: In formula (14), Σ s,m,i'→i,hefj (s) has been obtained in the continuous energy neutron transport calculation, and the scattering source in the i-th energy band is calculated by equation (14); Step 5: Assume that the total cross-section of the nuclear device is constant in the two-dimensional space region h, and assume that the neutron source is also constant in the two-dimensional space region h, and the flux is isotropic, then we get: in: Q in',hefj (s,Ω')—the total neutron source term with the order n of the expansion of the i-th energy segment in the neutron flight direction Ω', located at s, in the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction in the two-dimensional space region h; Formula (11) is combined into a matrix form and expressed as: make: Where: Q i,hefj —The total neutron source term of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial direction in the two-dimensional space region h; From formula (14), we can see that formula (12) can be written as: Where: Φ i,hefj —The neutron flux density of the ith energy segment in the jth characteristic line along the polar angle of the eth polar angle direction in the fth radial angle direction in the two-dimensional space region h; Σ t,i,hefj —The total cross section of the i-th energy segment in the j-th characteristic line along the polar angle of the e-th polar angle direction in the f-th radial direction in the two-dimensional space region h; Calculate the matrix Σ through matrix similarity transformation t,i,hefj All eigenvalues ​​of and its eigenvector And construct the matrix: For the matrix Σ t,i,hefj Perform a similarity transformation: Where: Λ t,i,hefj —In the two-dimensional space region h, along the j-th characteristic line of the polar angle in the f-th radial direction and the e-th polar angle direction, the total cross-sectional matrix Σ t,i,hefj A diagonal matrix composed of the eigenvalues ​​of ; B t,i,hefj —In the two-dimensional space region h, along the jth characteristic line of the polar angle in the fth angular direction and the eth polar angle direction, N i The total cross-section matrix Σ consists of linearly independent eigenvectors t,i,hefj Reversible matrices for similarity transformations; λ n —Total cross-section matrix Σ t,i,hefj The nth eigenvalue of ; ν n —Total cross-section matrix Σ t,i,hefj The nth linearly independent eigenvector of ; After sorting, we can get: Where: Y(s)—total cross-section matrix Σ t,i,hefj The product of the reversible matrix of similarity transformation and the flux moment vector; y n (s)—the nth value of Y(s); The general solution of equation (23) is obtained to obtain the outgoing neutron flux density: Integrate equation (25) along the characteristic line segment and arrange it to obtain the expression of average neutron angular flux density: The average flux of all characteristic line segments in the f direction in the two-dimensional space region h is summed according to the volume weight to obtain the average angular flux in the f direction in the two-dimensional space region h: Where: s h,f,j —The length of the jth characteristic line along the fth angular direction in the two-dimensional space region h; δA f — cross-sectional area of ​​the characteristic line in the f direction; V h —the volume of a two-dimensional space region h; —In the two-dimensional space region h, the nth order average angular flux on the ith energy segment of the jth characteristic line along the fth angular direction In actual calculation, it is necessary to correct the length of the characteristic line segment, that is: After obtaining the average angular flux of the polar angle e and the argument angle f in the two-dimensional space region h of the nuclear device by formula (27), the neutron standard flux density in the two-dimensional space region h is obtained by summing the angle weights of the argument angle and the polar angle of the neutron angular flux density: ω e —Polar angle direction weight; ω f —Argument weight; Then a characteristic line scan is performed to calculate the neutron flux density; Step 6: Determine whether the i-th energy segment flux of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device is converged. If not, return to step 4; Determine whether all energy segments of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device have been scanned, and if not, return to step 4; Determine whether the fluxes of each energy segment of the j-th characteristic line along the f-th direction in the two-dimensional space region h of the nuclear device are converged. If not, return to step 4; Step 7: Calculate the effective value-added factor k of the nuclear facility eff , and determine k eff Whether it converges. If not, return to step 4.

2. The method for calculating neutron and photon transport of continuous energy characteristic lines applied to nuclear-related devices according to claim 1 is characterized in that: The nuclear-related devices include reactor cores, radiation shielding layers, and medical radiation equipment.

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