CN114912064A - A Calculation Method of Continuous Energy Deterministic Neutron Transport Based on Function Expansion - Google Patents

Abstract

The invention discloses a neutron transport calculation method based on function expansion continuous energy determinism, which divides an energy region into a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region according to the characteristics of a microscopic cross section of nuclear reaction; each energy interval is divided into a plurality of energy sections; selecting different basis functions in different energy intervals as required, performing function expansion on the neutron angular flux density in different energy intervals, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space; solving the equation in different energy intervals to obtain neutron angular flux density coefficient moment, neutron source strong expansion coefficient moment and neutron angular flux density function of each energy section in the distinguishable resonance energy interval, further realizing the depiction of various nuclear reaction rates of various nuclides, and quantitatively describing the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity, the irradiation environment parameters and the like.

Description

A Calculation Method of Continuous Energy Deterministic Neutron Transport Based on Function Expansion

technical field

The invention relates to the field of neutron transport process simulation calculation and reactor physical analysis, in particular to a continuous energy deterministic neutron transport calculation method.

Background technique

A fission nuclear reactor is a device for controllable chain fission reactions, which can provide energy, radioisotopes, neutron beams, irradiation environment, etc. for industrial production and scientific research. In the active region of the core, in order to describe the transport process of neutrons in the medium, it is necessary to solve the neutron transport equation. This mathematical-physical equation is a differential-integral equation of the neutron angular flux density in a total of six-dimensional phase spaces such as three-dimensional space, one-dimensional neutron energy and two-dimensional flight direction, and belongs to the linear Boltzmann transport equation. The analytical solution of this equation can only be obtained in a very simplified case, which has important theoretical analysis value, but has no engineering practical significance. Therefore, nuclear reactor core physical engineering calculations are generally completed by computers using numerical calculation methods. At present, the numerical solution of the neutron transport equation is mainly divided into two categories. One is the deterministic method based on grid division and function expansion. The average neutron flux density in a finite number of phase space grids is used to approximate the The neutron flux density distribution function of continuous distribution in practical physical problems; the other is the probability theory method based on random sampling, also known as the Monte Carlo method, which converts the neutron flux density into the integral form of the statistical expectation of the random process, Through a large number of random sampling and statistical estimation, the estimated value of the neutron flux density distribution function in practical physical problems is obtained.

For the Monte Carlo method, credible results can be given if and only when the number of sample contributions of the target physical quantity is sufficient; especially for distributed physical quantities, the computational cost will increase sharply; in addition, the fine distribution of large-scale problems will be calculated. In the case of physical quantities, the method also faces the possible pseudo-convergence or even non-convergence problem in theory. Therefore, it is not suitable for massive engineering calculations under the structural geometry. In the deterministic method, the computational efficiency of the structural geometry is high, and for the joint phase space of the space of the neutron transport equation and the angle, very mature numerical discrete and iterative solution techniques have been developed, in which the approximation has become more and more. less. However, in sharp contrast to the spatial and angular phase space, the neutron energy phase space is still approximated by the traditional multi-group method. In the multi-group approximation process, the entire neutron energy range can be divided into several (G) intervals, which are called G energy groups; in each energy group, it is considered that the probability of all neutrons interacting with the nucleus is the same, It is called the average microscopic cross-section of the energy group; it is necessary to obtain the average microscopic cross-section of the non-resonant energy group based on the approximate energy spectrum and the evaluation core database, and then perform the resonance self-screening calculation in the resonance energy group to obtain the average microscopic cross-section of the resonant energy group. , then the real neutron transport simulation calculation can be carried out. This method discretizes the original equations containing continuous energy into G energy group equations through the multi-group approximation method. The number depends on the nature of the research problem and the precision requirements.

However, the simulation of neutron transport process based on the multi-group approximation technique faces a series of problems. On the one hand, the production of a multi-group database requires a pre-determined typical neutron energy spectrum, which cannot be related to specific problems; The scope of application and correction effect are very limited, which severely limits the design and analysis of nuclear energy installations.

In order to completely eliminate the problems caused by the multi-group approximation and improve the calculation accuracy and efficiency, the present invention directly starts from the continuous energy kernel database, and proposes a continuous energy deterministic neutron transport calculation method based on function expansion. Fundamentally eliminates multi-group database creation and resonance calculation operations, and provides more accurate data support for the design and analysis of nuclear energy installations.

SUMMARY OF THE INVENTION

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 deterministic neutron transport calculation method based on function expansion, which can directly and fundamentally eliminate the multi-group database creation and resonance calculation operations, and is a new method for nuclear energy. The design analysis of the device provides more accurate data support.

In order to achieve the above-mentioned purpose, the present invention adopts the following calculation scheme to implement:

A continuous energy deterministic neutron transport calculation method based on function expansion, the steps are as follows:

Step 1: Divide the energy region into thermal neutron energy region, discernible resonance energy region, indistinguishable resonance energy region and fast neutron energy region according to the micro-section characteristics of nuclear reaction; each energy region can be divided into several energies The denser the division of the energy segment, the higher the calculation accuracy;

Step 2: Select different basis functions R _n (u), n=0, 1, 2, ..., N in different energy intervals as required, and perform function expansion on the neutron angular flux density in different energy intervals , the neutron angular flux density is expanded into the form of the sum of the products of unknown coefficients and different order basis functions, as follows:

φ(r,Ω,u)--the neutron angular flux density (cm ^-2 s ^-1 ) flying along the angle Ω at the space r and the logarithmic energy of the flight energy is u; where u=lnE, E is The flight energy of the outgoing neutron, in MeV;

ψ _n (r,Ω)--the unknown expansion coefficient moment of the neutron angular flux density to be obtained;

The basis function R _n (u) selected in different energy intervals can accurately describe the fluctuation of neutron angular flux density with energy in this energy interval; characterization accuracy;

Step 3: Corresponding to the fast neutron energy region, the microscopic cross-section of the nuclear reaction changes continuously and smoothly with the incident neutron energy, so that the corresponding neutron angular flux density also presents a trend of continuous and smooth change; according to this feature, a smoothly changing basis function is selected , only considering the self-scattering source term in the intrinsic energy region (no scattering source term in the higher energy region to the lower energy region), through the basis function expansion of the neutron angular flux density in the fast neutron energy region, the energy phase space relative to the The equation of neutron angular flux density is converted into the equation of neutron angular flux density expansion coefficient moment in frequency domain space. That is, the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy segment in the fast neutron energy range are obtained; it avoids the situation that a preset energy spectrum is required for calculation in the multi-group method, and makes the calculation more accurate;

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u; σ _s,k (u'→u,Ω'→Ω)- - The k-th nuclide undergoes a scattering nuclear reaction with neutrons with logarithmic energy u' and flight direction Ω', and produces a microscopic scattering cross section (cm ² ) of neutrons with logarithmic energy u and flight direction Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

Step 4: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the indistinguishable resonance energy region can be obtained as,

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

Then it is necessary to consider the self-scattering source term in the indistinguishable resonance energy region. In the indistinguishable resonance energy region, the excitation curve of the microscopic cross-section changing with the incident neutron energy has excessively dense resonance peaks, and the specified incident neutron energy cannot be determined experimentally. The determined value of the micro-section can only give its probability density within a certain value range, which is called the probability measure, and the probability measure is expressed in the form of a probability table; basis function, and then expand the basis function of the neutron angular flux density in the indistinguishable resonance energy region, and convert the equation about the neutron angular flux density in the energy phase space into the neutron angular flux density in the frequency domain space. The equation of the expansion coefficient moment, the neutron transport equation format after the expansion of the energy region function is formula (2), and solving this equation can obtain the neutron angular flux density coefficient moment and neutron angular flux density coefficient moment of each energy segment in the indistinguishable resonance energy region. Source intensity expansion coefficient moment and neutron angle flux density function;

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

Step 5: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the discernible resonance energy region is:

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₃ ---Distinguishable resonance energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the indistinguishable resonance energy region calculated in step 4, the lower scattering source term from the indiscernible resonance energy region to the discernible resonance energy region can be obtained as,

R ² _n (u)--the basis function of the I ₂ area;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

In addition, the upper scattering source term from the thermal neutron energy region to the discernible resonance energy region is also required. This item needs to provide an initial value during the initial calculation, and then perform the upper scattering iteration with step 6 to obtain the updated upper scattering source term.

R ⁴ _n (u)--the basis function of the _14th region;

I ₄ -- thermal neutron energy interval;

I ₃ ---Distinguishable resonance energy interval;

In the distinguishable resonance energy region, the nuclear reaction cross-section also has a large number of resonance peaks. The closer to the indiscernible resonance energy region, the higher the density; the closer to the thermal neutron energy region, the wider the resonance peak, and the larger the affected energy range; The nuclear reaction cross section is represented in the form of a continuous function in the nuclear database; for this feature, a basis function that can accurately represent the abrupt change of the neutron flux density is used, and then the basis function for the neutron angular flux density in the discernible resonance energy region is used. Expand, convert the equation about neutron angular flux density in energy phase space into the equation of neutron angular flux density expansion coefficient moment in frequency domain space; The formula of the equation is formula (3), and by solving this equation, the neutron angular flux density coefficient moment, the neutron source intensity expansion coefficient moment and the neutron angular flux density function of each energy segment in the discriminable resonance energy interval are obtained;

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

R ³ _n (u)--the basis function of the _13th region;

R ⁴ _n (u)--the basis function of the _14th region;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

Step 6: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the thermal neutron energy region can be obtained as,

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₄ -- thermal neutron energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the indistinguishable resonance energy region calculated in step 4, the lower scattering source term from the indiscernible resonance energy region to the thermal neutron energy region can be obtained as,

R ² _n (u)--the basis function of the I ₂ area;

I ₂ --indiscernible resonance energy interval;

I ₄ -- thermal neutron energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the discriminable resonance energy range calculated in step 5, the lower scattering source term from the discriminable resonance energy range to the thermal neutron energy range can be obtained,

R ³ _n (u)--the basis function of the _13th region;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

In the thermal neutron energy region, the microscopic cross-section of the nuclear reaction changes continuously and smoothly with the incident neutron energy, so that the corresponding neutron flux density also presents a trend of continuous and smooth change; according to this characteristic, a smooth basis function is selected, and then the The basis function expansion of neutron angular flux density is carried out, and the equation about neutron angular flux density in energy phase space is converted into the equation of neutron angular flux density expansion coefficient moment in frequency domain space. The format of the neutron transport equation after the expansion of the basis function of each energy segment in this energy region is formula (4). Solving this equation can obtain the neutron angular flux density coefficient moment and neutron source intensity of each energy segment in the discernible resonance energy region. expansion coefficient moment and neutron angular flux density function,

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

R ³ _n (u)--the basis function of the _13th region;

R ⁴ _n (u)--the basis function of the _14th region;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

Step 7: Return to Step 5 to form an upper scattering iteration, and then check the relative error of the obtained neutron angular flux density coefficient moment in each energy interval before and after each iteration, and set the error limit according to the specific problem. If the error limit is exceeded, the calculation is judged to be converged, and the neutron angular flux expansion coefficient moment of each energy region is output; if the relative error exceeds the error limit, it is judged not to converge, and the process from step 5 to step 6 is continued until the set calculation iteration is reached. frequency;

Step 8: Using the obtained neutron angular flux expansion coefficient moments and the basis functions of the selected regions in different energy intervals, the neutron angular flux density distribution of continuous energy in all energy intervals can be obtained, and then the neutron angular flux density distribution of various nuclides can be obtained. Characterization of nuclear reaction rate, quantitative description of fission energy release, nuclide consumption and production, neutron beam intensity and irradiation environment parameters.

Aiming at the characteristics of the nuclear reaction micro-section in the non-resonant energy region, the indistinguishable resonance energy region and the discernible resonance energy region, the present invention proposes a targeted function expansion technology to properly describe the network coupling characteristics of neutrons in the energy space, and for continuous The neutron transport equation of energy is numerically discretized, and the solution of the neutron flux density distribution is transformed into the solution of the corresponding expansion coefficient moment, and the continuous distribution of physical quantities such as neutron flux density and nuclear reaction rate with neutron energy is obtained. Furthermore, the characterization of various nuclear reaction rates of various nuclides can be realized, and data support is provided for further quantitative description of the release of fission energy, the consumption and production of nuclides, the intensity of neutron beams, and the parameters of irradiation environment. The outstanding advantages of this method include: 1) eliminating the multi-group database production process and avoiding the problems caused by using typical neutron energy spectra in advance; The self-screening and mutual-screening effects in composition are fixed. 3) It can ensure both the calculation efficiency and the calculation accuracy.

Description of drawings

Figure 1 is a schematic diagram of the continuum energy deterministic neutron transport calculation based on function expansion.

Fig. 2 is a flow chart of the calculation method of continuous energy deterministic neutron transport based on function expansion.

Detailed ways

The present invention will be described in further detail below in conjunction with the accompanying drawings and specific embodiments:

随能量的变化，其中实线表示需要求解的真实的中子角通量密度

虚线表示实际计算求解的中子角通量密度

As shown in Figure 1, the curve below the coordinate axis represents the neutron angular flux density at the position r in any energy interval and the flight angle is Ω

as a function of energy, where the solid line represents the true neutron angular flux density to be solved for

The dotted line represents the actual calculated neutron angular flux density

The method principle of the present invention is:

According to the micro-section characteristics of nuclear reaction, the energy area is divided into: thermal neutron energy area, discernible resonance energy area, indistinguishable resonance energy area, fast neutron energy area; each energy area can be divided into several energy segments. The neutron angular flux density has a strong correlation with the nuclear reaction cross section, so different basis functions R _n (u), n = 0, 1, 2, ..., N, are selected in different energy intervals according to the needs. The neutron angular flux density in the energy interval is functionally expanded, and the neutron angular flux density is expanded into the form of the sum of the products of unknown coefficients and different order basis functions, and the relationship between the neutron angular flux density in the energy phase space is Equation, converted into the equation of the coefficient moment of the neutron angular flux density expansion in the frequency domain space. Solving this equation in different energy intervals can obtain the neutron angular flux density coefficient moment, neutron source intensity expansion coefficient moment and neutron angular flux density function of each energy segment in the discernible resonance energy interval, and then realize the Characterization of various nuclear reaction rates of nuclides, quantitative description of fission energy release, nuclide consumption and production, neutron beam intensity, irradiation environmental parameters, etc.

As shown in Figure 2, the specific implementation steps of the present invention are as follows:

A continuous energy deterministic neutron transport calculation method based on function expansion, the steps are as follows:

Step 1: Divide the energy region into thermal neutron energy region, discernible resonance energy region, indistinguishable resonance energy region and fast neutron energy region according to the micro-section characteristics of nuclear reaction; each energy region can be divided into several energies The denser the division of the energy segment, the higher the calculation accuracy;

Step 2: Select different basis functions R _n (u), n=0, 1, 2, ..., N in different energy intervals as required, and perform function expansion on the neutron angular flux density in different energy intervals , the neutron angular flux density is expanded into the form of the sum of the products of unknown coefficients and different order basis functions, as follows:

--在空间r处沿角度Ω飞行且飞行能量的对数能量为u的中子角通量密度(cm^-2s^-1)；其中u＝lnE，E为出射中子飞行能量，单位为：MeV；

--The neutron angular flux density (cm ^-2 s ^-1 ) flying along the angle Ω at the space r and the logarithmic energy of the flight energy is u; where u=lnE, E is the flight energy of the outgoing neutron, in units of : MeV;

ψ _n (r,Ω)--the unknown expansion coefficient moment of the neutron angular flux density to be obtained;

The basis function R _n (u) selected in different energy intervals can accurately describe the fluctuation of neutron angular flux density with energy in this energy interval; characterization accuracy;

Step 3: Corresponding to the fast neutron energy region, the microscopic cross-section of the nuclear reaction changes continuously and smoothly with the incident neutron energy, so that the corresponding neutron angular flux density also presents a trend of continuous and smooth change; according to this feature, a smoothly changing basis function is selected , only considering the self-scattering source term in the intrinsic energy region (no scattering source term in the higher energy region to the lower energy region), through the basis function expansion of the neutron angular flux density in the fast neutron energy region, the energy phase space relative to the The equation of neutron angular flux density is converted into the equation of neutron angular flux density expansion coefficient moment in frequency domain space. That is, the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy segment in the fast neutron energy range are obtained; it avoids the situation that a preset energy spectrum is required for calculation in the multi-group method, and makes the calculation more accurate;

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

Step 4: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the indistinguishable resonance energy region can be obtained as,

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

Then it is necessary to consider the self-scattering source term in the indistinguishable resonance energy region. In the indistinguishable resonance energy region, the excitation curve of the microscopic cross-section changing with the incident neutron energy has excessively dense resonance peaks, and the specified incident neutron energy cannot be determined experimentally. The determined value of the micro-section can only give its probability density within a certain value range, which is called the probability measure, and the probability measure is expressed in the form of a probability table; basis function, and then expand the basis function of the neutron angular flux density in the indistinguishable resonance energy region, and convert the equation about the neutron angular flux density in the energy phase space into the neutron angular flux density in the frequency domain space. The equation of the expansion coefficient moment, the neutron transport equation format after the expansion of the energy region function is formula (2), and solving this equation can obtain the neutron angular flux density coefficient moment and neutron angular flux density coefficient moment of each energy segment in the indistinguishable resonance energy region. Source intensity expansion coefficient moment and neutron angle flux density function;

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

Step 5: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the discernible resonance energy region is:

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₃ ---Distinguishable resonance energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the indistinguishable resonance energy region calculated in step 4, the lower scattering source term from the indiscernible resonance energy region to the discernible resonance energy region can be obtained as,

R ² _n (u)--the basis function of the I ₂ area;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

In addition, the upper scattering source term from the thermal neutron energy region to the discernible resonance energy region is also required. This item needs to provide an initial value during the initial calculation, and then perform the upper scattering iteration with step 6 to obtain the updated upper scattering source term.

R ⁴ _n (u)--the basis function of the _14th region;

I ₄ -- thermal neutron energy interval;

I ₃ ---Distinguishable resonance energy interval;

In the distinguishable resonance energy region, the nuclear reaction cross-section also has a large number of resonance peaks. The closer to the indiscernible resonance energy region, the higher the density; the closer to the thermal neutron energy region, the wider the resonance peak, and the larger the affected energy range; The nuclear reaction cross section is represented in the form of a continuous function in the nuclear database; for this feature, a basis function that can accurately represent the abrupt change of the neutron flux density is used, and then the basis function for the neutron angular flux density in the discernible resonance energy region is used. Expand, convert the equation about neutron angular flux density in energy phase space into the equation of neutron angular flux density expansion coefficient moment in frequency domain space; The formula of the equation is formula (3), and by solving this equation, the neutron angular flux density coefficient moment, the neutron source intensity expansion coefficient moment and the neutron angular flux density function of each energy segment in the discriminable resonance energy interval are obtained;

In this step, on the one hand, the resonance calculation with a large amount of calculation is eliminated, which greatly reduces the requirements for computing resources in engineering calculations. The calculation is more accurate.

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

R ³ _n (u)--the basis function of the _13th region;

R ⁴ _n (u)--the basis function of the _14th region;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

Step 6: Using the neutron angular flux density coefficient moments of each energy segment in the fast neutron energy region calculated in step 3, the lower scattering source term from the fast neutron energy region to the thermal neutron energy region can be obtained as,

R ¹ _n (u)--the basis function of the I ₁ area;

I ₁ --fast neutron energy interval;

I ₄ -- thermal neutron energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the indistinguishable resonance energy region calculated in step 4, the lower scattering source term from the indiscernible resonance energy region to the thermal neutron energy region can be obtained as,

R ² _n (u)--the basis function of the I ₂ area;

I ₂ --indiscernible resonance energy interval;

I ₄ -- thermal neutron energy interval;

Using the neutron angular flux density coefficient moments of each energy segment in the discriminable resonance energy range calculated in step 5, the lower scattering source term from the discriminable resonance energy range to the thermal neutron energy range can be obtained,

R ³ _n (u)--the basis function of the _13th region;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

In the thermal neutron energy region, the microscopic cross-section of the nuclear reaction changes continuously and smoothly with the incident neutron energy, so that the corresponding neutron flux density also presents a trend of continuous and smooth change; according to this characteristic, a smooth basis function is selected, and then the The basis function expansion of neutron angular flux density is carried out, and the equation about neutron angular flux density in energy phase space is converted into the equation of neutron angular flux density expansion coefficient moment in frequency domain space. The format of the neutron transport equation after the expansion of the basis function of each energy segment in this energy region is formula (4). Solving this equation can obtain the neutron angular flux density coefficient moment and neutron source intensity of each energy segment in the discernible resonance energy region. expansion coefficient moment and neutron angular flux density function,

where:

Ω--the angle of outgoing neutrons in space;

Ω'--the incident neutron angle in space;

r--spatial position;

u – logarithmic energy of the outgoing neutron energy;

u’ – logarithmic energy of incident neutron energy;

N _k (r)--nucleus density of the kth nuclide at space r (cm ^-3 );

σ _t,k (u)--the total microscopic cross-section (cm ² ) of the nuclear reaction between the k-th nuclide and the neutron with logarithmic energy u;

σ _s,k (u'→u,Ω'→Ω)--the k-th nuclide undergoes a nuclear scattering reaction with the neutron with logarithmic energy u' and flight direction Ω' and produces logarithmic energy u, flight direction The microscopic scattering cross section (cm ² ) of neutrons in the direction of Ω;

q _n (r,Ω)--unknown expansion coefficient moment of neutron source intensity to be determined;

R ¹ _n (u)--the basis function of the I ₁ area;

R ² _n (u)--the basis function of the I ₂ area;

R ³ _n (u)--the basis function of the _13th region;

R ⁴ _n (u)--the basis function of the _14th region;

I ₁ --fast neutron energy interval;

I ₂ --indiscernible resonance energy interval;

I ₃ ---Distinguishable resonance energy interval;

I ₄ -- thermal neutron energy interval;

Step 7: Return to Step 5 to form an upper scattering iteration, and then check the relative error of the obtained neutron angular flux density coefficient moment in each energy interval before and after each iteration, and set the error limit according to the specific problem. If the error limit is exceeded, the calculation is judged to be converged, and the neutron angular flux expansion coefficient moment of each energy region is output; if the relative error exceeds the error limit, it is judged not to converge, and the process from step 5 to step 6 is continued until the set calculation iteration is reached. frequency;

Step 8: Using the obtained neutron angular flux expansion coefficient moments and the basis functions of the selected regions in different energy intervals, the neutron angular flux density distribution of continuous energy in all energy intervals can be obtained, and then the neutron angular flux density distribution of various nuclides can be obtained. Characterization of nuclear reaction rate, quantitative description of fission energy release, nuclide consumption and production, neutron beam intensity and irradiation environment parameters.

According to the characteristics of the nuclear reaction micro-section in the non-resonant energy region, the indistinguishable resonance energy region and the discernible resonance energy region, a targeted function expansion technique is proposed to properly characterize the network coupling characteristics of neutrons in the energy space, and for the continuous energy The neutron transport equation is numerically discretized, and the solution of the neutron flux density distribution is transformed into the solution of the corresponding expansion coefficient moment, and the continuous distribution of physical quantities such as neutron flux density and nuclear reaction rate with neutron energy is obtained. Furthermore, the characterization of various nuclear reaction rates of various nuclides can be realized, and data support is provided for further quantitative description of the release of fission energy, the consumption and production of nuclides, the intensity of neutron beams, and the parameters of irradiation environment. The outstanding advantages of this method include: 1) eliminating the multi-group database production process and avoiding the problems caused by using typical neutron energy spectra in advance; The self-screening and mutual-screening effects in composition are fixed. 3) It can ensure both the calculation efficiency and the calculation accuracy.

On the one hand, this method is mainly completed by computer, so multiple simulations can be performed simultaneously under typical working conditions according to the design scheme, which greatly reduces the time cost; Therefore, using this method can reduce the number of experimental analysis and the huge experimental cost under the premise of ensuring a certain calculation accuracy. It can also provide design ideas for the design of related nuclear energy devices, and provide quantitative data for further improving the performance of nuclear energy devices.

Claims (1)

1. A neutron transport calculation method based on a function expansion continuous energy determinism is characterized in that: the method comprises the following steps:

step 1: the energy region is divided into the following parts according to the microscopic cross section characteristics of nuclear reaction: a thermal neutron energy region, a distinguishable resonance energy region, an indistinguishable resonance energy region and a fast neutron energy region; each energy interval can be divided into a plurality of energy sections;

step 2: selecting different basis functions R according to requirements in different energy intervals _n (u), N being 0,1,2, …, N, and developing a function of the neutron angular flux density over different energy intervals, the neutron angular flux density being developed as a sum of products of unknown coefficients and different order basis functions, as follows:

phi (r, omega, u) -the neutron angular flux density in space r along an angle omega with logarithmic energy of flight energy u in units of: cm of ^-2 s ^-1 (ii) a Wherein u is lnE, and E is the flight energy of the emergent neutron, and the unit is: MeV;

ψ _n (r, Ω) -unknown expansion coefficient moment of neutron angular flux density to be solved;

selected basis functions R in different energy intervals _n (u) accurately depicting the variation of neutron angular flux density with energy fluctuation within the energy interval; the greater the expansion order N of the basis function is, the more the description precision of different energy regions can be met;

and step 3: selecting a smoothly-changed basis function, only considering a self-scattering source item of an instron, performing basis function expansion on neutron angular flux density of a fast neutron energy area, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to a neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy area function expansion is in a format of a formula (1), and solving the equation to obtain the neutron angular flux density coefficient moment and the neutron angular flux density function of each energy section of a fast neutron energy interval;

in the formula:

omega-angle of neutron extraction in space;

Ω' - -angle of incident neutrons in space;

r- -spatial position;

u-log energy of the emitted neutron energy;

u' -log energy of incident neutron energy;

N _k (r) -nuclear density of the kth nuclear species in space r in units of: cm of ^-3 ；

σ _t,k Microscopic total cross-section of nuclear reaction of (u) -kth nuclide with a neutron of logarithmic energy u in units of: cm of ² ；

σ _s,k (u '→ u, Ω' → Ω) - -kth nuclide scatters nuclei with neutrons having a logarithmic energy u 'and a flight direction Ω' and produces microscopic scattering cross sections with neutrons having a logarithmic energy u and a flight direction Ω, in units of: cm ² ；

q _n (r, omega) -unknown expansion coefficient moment of neutron source intensity to be solved;

R ¹ _n (u) - - - (I) th ₁ A base function of the region;

I ₁ -a fast neutron energy interval;

and 4, step 4: obtaining the lower scattering source term from the fast neutron energy region to the indistinguishable resonance energy region by using the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,

R ¹ _n (u) - - - (I) th ₁ Basis functions of regions；

I ₁ -fast neutron energy interval;

I ₂ -an indistinguishable resonance energy interval;

then, self-scattering source items in an indistinguishable resonance energy interval need to be considered, in an indistinguishable resonance energy area, excitation curves of microscopic sections changing along with incident neutron energy have excessively dense formants, the determination value of the specified incident neutron energy microscopic section cannot be determined experimentally, only the probability density of the specified incident neutron energy microscopic section in a certain value range can be given, the probability density is called probability measure, and the probability measure is expressed in the form of a probability table; using a basis function which can accurately measure the distribution of probability while considering the energy distribution, then performing basis function expansion on the neutron angular flux density of an indistinguishable resonance energy region, converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space, wherein the neutron transport equation after the energy region function expansion is in a formula (2), and solving the equation to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the indistinguishable resonance energy region;

in the formula:

R ² _n (u) - - - (I) th ₂ A base function of the region;

I ₂ -an indistinguishable resonance energy interval;

and 5: obtaining a lower scattering source term from the fast neutron energy region to the distinguishable resonance energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,

I ₃ -distinguishable resonance energy interval;

the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy interval obtained by calculation in the step 4 is utilized to obtain a lower scattering source item from the indistinguishable resonance energy area to the distinguishable resonance energy area,

in addition, the neutron energy region is heated to the upper scattering source item of the distinguishable resonance energy region, the upper scattering source item needs to provide an initial value during initial calculation, then the upper scattering iteration is carried out with the step 6 to obtain an updated upper scattering source item,

R ⁴ _n (u) - - - (I) th ₄ A base function of the region;

I ₄ -a thermal neutron energy interval;

using a basis function capable of accurately representing the abrupt change of the neutron flux density, then performing basis function expansion on the neutron angular flux density of a distinguishable resonance energy region, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the expansion coefficient moment of the neutron angular flux density in a frequency domain space; the format of a neutron transport equation after the expansion of the basis functions of each energy section of the energy zone is a formula (3), and the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the distinguishable resonance energy zone;

in the formula:

R ³ _n (u) - - - (I) th ₃ A base function of the region;

step 6: obtaining a lower scattering source term from the fast neutron energy region to the thermal neutron energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the fast neutron energy region calculated in the step 3,

obtaining the lower scattering source term from the indistinguishable resonance energy region to the thermal neutron energy region by utilizing the neutron angular flux density coefficient moment of each energy section of the indistinguishable resonance energy region obtained by calculation in the step 4,

obtaining a lower scattering source term from the distinguishable resonance energy region to the thermal neutron energy region by using the neutron angular flux density coefficient moment of each energy section of the distinguishable resonance energy region calculated in the step 5,

selecting a smooth basis function, then performing basis function expansion on the neutron angular flux density of the thermal energy region, and converting an equation related to the neutron angular flux density in an energy phase space into an equation related to the neutron angular flux density expansion coefficient moment in a frequency domain space. The format of the neutron transport equation after the basis functions of each energy section of the energy zone are expanded is formula (4), the equation is solved to obtain the neutron angular flux density coefficient moment, the neutron source strong expansion coefficient moment and the neutron angular flux density function of each energy section of the identifiable resonance energy zone,

and 7: returning to the step 5, forming an upper scattering iteration, checking the relative error of the neutron angular flux density coefficient moment of each energy interval obtained by solving before and after each iteration, setting an error limit according to a specific problem, judging calculation convergence if the relative error does not exceed the error limit, and outputting the neutron angular flux expansion coefficient moment of each energy zone; if the relative error exceeds the error limit, judging that the convergence is not achieved, and continuing the process from the step 5 to the step 6 until the set calculation iteration number is reached;

and 8: the neutron angular flux expansion coefficient moment and the basis function of the selected area in different energy intervals are utilized to obtain the neutron angular flux density distribution of continuous energy in all the energy intervals, so that the depiction of various nuclear reaction rates of various nuclides is realized, and the release of fission energy, the consumption and production of the nuclides, the neutron beam intensity and the irradiation environment parameters are quantitatively described.

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