Rapid calculation method for effective proliferation factors of spent fuel pool under emergency working condition based on neutron diffusion variational nodal method

Abstract

The invention relates to the technical field of nuclear and radiation nuclear emergency, in particular to a spent fuel pool effective multiplication factor rapid calculation method based on a neutron diffusion variational nodal method under an emergency working condition. The method comprises the following specific steps: decomposing the spent fuel pool into a plurality of segments; establishing a functional on each segment; expanding a few group constant by using a fragment constant function; expanding the neutron-flux density and the source item by using a mixed function; establishing a response relational expression based on the previous steps; and calculating to obtain effective proliferation factors. The rapid spent fuel pool effective multiplication factor calculation method is adopted, the effective multiplication factors can be given in real time, the critical safety margin of the spent fuel pool under the nuclear emergency working condition is determined, a scientific and accurate basis is provided for emergency working condition evaluation and emergency decision, the pertinence of nuclear emergency response actions is improved, and the economic benefit is increased. And the method plays an important role in guaranteeing the safety and economy of nuclear energy.

Description

Technical Field

[0001] The present invention relates to the technical field of nuclear and radiation nuclear emergency, and particularly to a rapid calculation method for the effective multiplication factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method. Background Technique

[0002] The Fukushima nuclear accident in Japan in 2011 left an important warning: both the in-core fuel assemblies after reactor shutdown and the spent fuel assemblies stored in the spent fuel pool may face the safety risk of returning to criticality. The parameter directly quantitatively characterizing critical safety is the effective multiplication factor. The closer this parameter is to 1, the closer the system is to criticality, and the higher the critical safety risk.

[0003] Currently, relevant operating units mainly monitor whether the spent fuel pool reaches criticality through critical (neutron or γ) detectors under nuclear emergency conditions to judge the critical safety risk. They cannot obtain the effective multiplication factor in real time, and thus cannot clarify the critical safety margin when criticality has not occurred, adding additional uncertainties to the nuclear emergency condition evaluation and emergency decision-making for the spent fuel pool before criticality. In contrast, a rapid calculation method for the effective multiplication factor of the spent fuel pool can give the effective multiplication factor in real time, clarify the critical safety margin of the spent fuel pool under nuclear emergency conditions, provide a scientific and accurate basis for emergency condition evaluation and emergency decision-making, improve the pertinence of nuclear emergency response actions, and play an important role in ensuring the safety and economy of nuclear energy. Summary of the Invention

[0004] The object of the present invention is to address the problems in the background technique and propose a rapid calculation method for the effective multiplication factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method. In view of the actual need for rapid calculation of the effective multiplication factor of the spent fuel pool under nuclear emergency conditions, a neutron diffusion non-uniform variational nodal method that can explicitly handle the complex non-uniform distribution within the nodal blocks of the spent fuel pool under nuclear emergency conditions is established. Through the mixed basis function expansion technique that can accurately and efficiently describe the deformed neutron flux density distribution, rapid calculation of the effective multiplication factor of the spent fuel pool under nuclear emergency conditions is achieved while ensuring the calculation accuracy.

[0005] The technical solution of the present invention:

[0006] The first aspect of the present invention provides a rapid calculation method for the effective multiplication factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method, including the following specific steps:

[0007] S1. Decompose the spent fuel pool into a number of nodal blocks;

[0008] S2. Establish a functional on each nodal block;

[0009] S3. Expand the few-group constants using piecewise constant functions;

[0010] S4. Expand the neutron flux density and source term using a hybrid function;

[0011] S5. Establish a response relationship based on steps S3 and S4;

[0012] S6. Calculate the effective multiplication factor.

[0013] Preferably, in step S1, the non-uniform variational nodal method is adopted to discretize the neutron flux density, neutron source term, and neutron current density of the components in the spent fuel pool, for dealing with the complex non-uniform distribution caused by the dumping of components.

[0014] Preferably, in step S2, the functional discretized form that can handle complex non-uniform distributions is derived based on the neutron diffusion equation and its boundary conditions;

[0015]

[0016] Φ g (r) - 2J γ,g (r) = β γ,g ·(Φ g (r) + 2J γ,g (r)) r ∈ Γ γ

[0017] where D g (r) —— the neutron diffusion coefficient of the g-th energy group / cm; Φ g (r) —— the neutron flux density / cm -2 ·s -1 ; Σ r,g —— the neutron removal cross section of the g-th energy group / cm -1 ; S g (r) represents the neutron source term (cm -2 s -1 ), including the scattered neutron source term and the fission neutron source term; Σ g'→g (r) —— the neutron scattering cross section from the g'-th energy group to the g-th energy group / cm -1 ; χ g —— the neutron fission spectrum of the g-th energy group; k eff —— the effective multiplication factor; ν —— the average number of neutrons produced per fission; Σ f,g' (r) —— the neutron fission cross section of the g-th energy group / cm -1 ; β γ,g —— the neutron albedo of the γ surface and the g-th energy group; —— the net neutron current density along the outer normal direction of the γ surface and the g-th energy group / cm -2 ·s -1 ; nγ —— the outer normal direction vector of the γ plane.

[0018] Preferably, according to the material distribution and geometric distribution of the spent fuel pool, the overall problem is divided into multiple nodal blocks, and the functional of the neutron flux density distribution is written in the form of the sum of local functionals of each nodal block;

[0019]

[0020] where v is the number of the nodal block; the single nodal block functional is:

[0021]

[0022] Discretize the functional of each nodal block:

[0023]

[0024] In the formula, f i (r), h γ,l (r) respectively represent the complete orthogonal polynomial basis functions defined inside and on the surface of the nodal block, and f, h γ respectively represent the vectors of the expansion basis functions inside and on the surface of the nodal block, s g and j γ,g respectively represent the vectors of the expansion moments of the neutron flux density, neutron source term, and neutron current density on the boundary surface of the nodal block.

[0025] Preferably, in step S3, discretize the functional of each nodal block and expand the few-group constants using piecewise constant functions to explicitly express the complex inhomogeneity of the material;

[0026]

[0027] Among them, respectively represent the expansion moments of the few-group constants and their vectors. In this way, the complex inhomogeneity of the material can be explicitly expressed in the discrete form of the functional.

[0028] Preferably, based on the variational principle, obtain the response relationship between the expansion moment vectors of each variable:

[0029]

[0030] In the formula, Ψ g represents the vector of the expansion moment of the neutron flux density on the surface of the nodal block, and the matrices A and M are determined by the geometry, material, and expansion basis functions of the nodal block. Define the expansion moment of the partial neutron current density on the surface of the nodal block:

[0031]

[0032] Substituting the above formula, a matrix equation representing the neutron balance relationship inside the node and the neutron continuity relationship on the node surface is obtained:

[0033]

[0034] Among them, the response matrices H, C, B, and R of the node are only related to the geometry and material properties; the matrix equations respectively include the neutron balance relationship inside the node and the neutron current continuity relationship on the node surface.

[0035] Preferably, orthogonal polynomials are used as the basis functions for expansion and discretization to obtain the matrix equations of the neutron balance relationship inside the node and the neutron current continuity relationship on the node surface.

[0036] Preferably, a mixed form of first-order hyperbolic functions and polynomial functions is used as the expansion basis functions for the neutron flux density and neutron source term inside the node to reduce the number of expansion basis functions and the consumption of computing resources.

[0037] Preferably, for normal fuel assemblies, a functional is established using uniform few-group constants, and for damaged or tilted fuel assemblies, a functional is established using piecewise functions according to the actual situation, and combined with establishing a global functional, an iterative calculation is performed using the iterative process of the traditional variational nodal method.

[0038] The second aspect of the present invention provides an application, using the above-mentioned rapid calculation method for the effective multiplication factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method to calculate the effective multiplication factor of the spent fuel pool under emergency conditions.

[0039] Compared with the prior art, the present invention has the following beneficial technical effects:

[0040] 1. In view of the actual demand for rapid calculation of the effective multiplication factor of a spent fuel pool under nuclear emergency conditions, the present invention establishes a neutron diffusion non-uniform variational nodal method that can explicitly handle the complex non-uniform distribution inside the nodes of the spent fuel pool under nuclear emergency conditions. Through the mixed basis function expansion technology that can accurately and efficiently describe the abnormal neutron flux density distribution, the rapid calculation of the effective multiplication factor of the spent fuel pool under nuclear emergency conditions is realized while ensuring the calculation accuracy.

[0041] 2. The present invention adopts a rapid calculation method for the effective multiplication factor of the spent fuel pool, which can give the effective multiplication factor in real time, clarify the critical safety margin of the spent fuel pool under nuclear emergency conditions, provide a scientific and accurate basis for emergency condition evaluation and emergency decision-making, improve the pertinence of nuclear emergency response actions, and play an important role in ensuring the safety and economy of nuclear energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic diagram of the system flow structure of an embodiment of the present invention. Detailed implementation mode

[0043] Embodiment 1

[0044] In the traditional variational nodal method, the neutron flux density, neutron source term inside the node and neutron current density on the node surface are discretized using orthogonal polynomials. Therefore, it can only handle nodes with a uniform material distribution in space. The neutron diffusion non-uniform variational nodal method represents the cross-section inside the node as a function of position and can describe non-uniform material distributions.

[0045] Due to various considerations such as application scenarios, computational efficiency, and programming difficulty, the existing neutron diffusion non-uniform variational nodal method can only handle axially gradually varying non-uniform distributions or piecewise uniform distributions. However, for the components in the spent fuel pool under the nuclear emergency conditions targeted by the present invention, changes such as tipping may occur, which results in a more complex non-uniform distribution inside the node. The existing neutron diffusion non-uniform variational nodal method cannot handle this non-uniform distribution.

[0046] Based on this, as Figure 1 shown, a rapid calculation method for the effective multiplication factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method proposed by the present invention includes the following specific steps:

[0047] S1. Decompose the spent fuel pool into several nodes;

[0048] S2. Establish a functional on each node;

[0049] S3. Expand the few-group constants using piecewise constant functions;

[0050] S4. Expand the neutron flux density and source term using hybrid functions;

[0051] S5. Establish a response relationship based on steps S3 and S4;

[0052] S6. Calculate the effective multiplication factor.

[0053] In order to handle the complex non-uniform distribution caused by the tipping of the component, starting from the neutron diffusion equation, a discrete form of the functional that can handle the complex non-uniform distribution is derived.

[0054] The non-uniform variational nodal method starts from the neutron diffusion equation and its boundary conditions:

[0055]

[0056] Φ g (r)-2J γ,g (r)=β γ,g ·(Φ g (r)+2J γ,g(r)) r ∈ Γ γ (3)

[0057] where D g (r) —— neutron diffusion coefficient of the g-th energy group / cm; Φ g (r) —— neutron flux density / cm -2 ·s -1 ; Σ r,g —— neutron removal cross section of the g-th energy group / cm -1 ; S g (r) represents the neutron source term (cm -2 s -1 ), including the scattered neutron source term and the fission neutron source term; Σ g'→g (r) —— neutron scattering cross section from the g'-th energy group to the g-th energy group / cm -1 ; χ g —— neutron fission spectrum of the g-th energy group; k eff —— effective multiplication factor; ν —— average number of neutrons produced per fission; Σ f,g' (r) —— neutron fission cross section of the g-th energy group / cm -1 ; β γ,g —— albedo of neutrons on the γ-th surface and in the g-th energy group; —— net neutron current density along the outer normal direction on the γ-th surface and in the g-th energy group / cm -2 ·s -1 ; n γ —— outer normal direction vector of the γ-th surface. Others are all common symbols in the field of neutron diffusion calculation.

[0058] According to the material distribution and geometric distribution of the spent fuel pool, the overall problem can be divided into a certain number of nodes, and the functional of the neutron flux density distribution of the whole problem can be written in the form of the sum of local functionals on each node:

[0059]

[0060] where v is the number of the node. The functional of a single node is:

[0061]

[0062] Discretize the functional of each node:

[0063]

[0064] In the formula, f i (r), h γ,l (r) represent the complete orthogonal polynomial basis functions defined inside and on the surface of the node respectively, and f, h γ respectively represent the vectors of the expansion basis functions inside and on the surface of the node, s g 、j γ,g respectively represent the vectors of the expansion moments of the neutron flux density, the neutron source term, and the neutron current density on the nodal interface.

[0065] In addition, according to the non-uniform structural characteristics of the spent fuel pool under nuclear emergency conditions, a piecewise constant function c k (t,z) (function vector c) is selected to expand the few-group constants.

[0066]

[0067] Among them, respectively represent the expansion moments of the few-group constants and their vectors. In this way, the complex non-uniformity of the material can be explicitly expressed in the discrete form of the functional.

[0068] According to the variational principle, the response relationship between the expansion moment vectors of each variable can be obtained:

[0069]

[0070] In the formula, Ψ g represents the vector of the expansion moment of the neutron flux density on the nodal surface. The matrices A and M are determined by the geometry, materials, and expansion basis functions of the node. Define the expansion moment of the partial neutron current density on the nodal surface:

[0071]

[0072] Substitute into Eqs. (8) and (9) to obtain the matrix equations representing the neutron balance relationship inside the node and the neutron continuity relationship on the nodal surface respectively:

[0073]

[0074] Among them, the response matrices H, C, B, and R of the node are only related to the geometry and material properties. The matrix equations (11) and (12) respectively contain the neutron balance relationship inside the node and the neutron current continuity relationship on the nodal surface. Select its iterative solution strategy to realize the non-uniform variational nodal method for neutron diffusion. Among them, the construction of the functional is based on strict three-dimensional diffusion theory, and the order of discretization of the unknowns is not limited, with broad applicability. And the complex non-uniformity of the materials that may occur under nuclear emergency conditions is also explicitly processed in the discrete functional by using the piecewise constant function c k (t,z) to expand the few-group constants.

[0075] As described above, the neutron flux density, neutron source term within the variational nodal method node, and neutron current density on the node surface are all expanded and discretized using orthogonal polynomials as basis functions. The analytical solution of the diffusion equation is in the form of hyperbolic functions, and hyperbolic functions are good at describing distributions with large gradients, which conform to the distorted distribution characteristics of the neutron flux density under non-uniform distributions. Moreover, the first-order hyperbolic function can maximize the display of large-gradient distributions; polynomial functions have the characteristics of wide adaptability, fast calculation speed, simple expression, etc., but expressing distributions with large gradients requires a relatively high expansion order. Therefore, in this patent, the analytical solution of the diffusion equation (using the first-order hyperbolic function) and polynomial functions are mixed as the expansion basis functions of the neutron flux density and neutron source term within the node and applied in the improved neutron diffusion variational nodal method, which can reduce the number of expansion basis functions as much as possible under the condition of meeting a certain accuracy, thereby reducing the consumption of computing resources.

[0076] For the normal fuel assemblies in the spent fuel pool under emergency conditions, functionals for each assembly are established using homogeneous few-group constants, and for the damaged or tilted fuel assemblies, functionals for their assemblies are established using piecewise functions according to the actual situation. Combining them together to establish a global functional and using the iterative process of the traditional variational nodal method for iteration can quickly calculate the effective multiplication factor of the spent fuel pool under emergency conditions.

[0077] Embodiment 2

[0078] This embodiment proposes an application that uses the rapid calculation method for the effective multiplication factor of the spent fuel pool under emergency conditions based on the neutron diffusion variational nodal method in Embodiment 1 to calculate the effective multiplication factor of the spent fuel pool under emergency conditions. Compared with the existing solutions, this embodiment adopts a rapid calculation method for the effective multiplication factor of the spent fuel pool, which can give the effective multiplication factor in real time, clarify the critical safety margin of the spent fuel pool under nuclear emergency conditions, provide a scientific and accurate basis for emergency condition evaluation and emergency decision-making, improve the pertinence of nuclear emergency response actions, and play an important role in ensuring the safety and economy of nuclear energy.

[0079] The embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings. However, the present invention is not limited thereto. Various changes can be made without departing from the spirit of the present invention within the knowledge scope of those skilled in the art to which the present invention pertains.

Claims

1. A method for quickly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method, characterized in that: The specific steps include: S1. Decompose the spent fuel pool into several sections; S2, establish functionals on each node; S3, use the piecewise constant function to expand the small group constant; S4. Expand the neutron flux density and source term using a hybrid function; S5, establishing a response relationship based on step S3 and step S4; S6. Calculate and obtain the effective proliferation factor.

2. According to claim 1, a method for quickly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method is characterized in that: Step S1 uses the non-uniform variational nodal method to discretize the neutron flux density, neutron source term and neutron current density of the components in the spent fuel pool, so as to deal with the complex non-uniform distribution caused by the dumping of the components.

3. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 2 is characterized in that: Step S2 derives a functional discrete form that can handle complex non-uniform distribution based on the neutron diffusion equation and its boundary conditions; Φ g (r)-2J γ,g (r)=β γ,g ·(Φ g (r)+2J γ,g (r))r∈Γ γ Among them, D g (r)——neutron diffusion coefficient of the g-th energy group / cm; Φ g (r)——neutron flux density / cm -2 ·s -1 ;Σ r,g ——Neutron removal cross section of the g-th energy group / cm -1 ; S g (r) represents the neutron source term (cm -2 s -1 ), including the scattered neutron source term and the fission neutron source term; Σ g'→g (r)——neutron scattering cross section from the g'th energy group to the gth energy group / cm -1 ; g —— Neutron fission spectrum of the g-th energy group; k eff ——effective multiplication factor; ν——average number of neutrons produced per fission; Σ f,g' (r)——neutron fission cross section of the g-th energy group / cm -1 β γ,g ——neutron albedo of the γth surface and the gth energy group; ——Net neutron flux density of the γth surface and the gth energy group along the external normal direction / cm -2 ·s -1 ;n γ ——The external normal direction vector of the γth surface.

4. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 3 is characterized in that: According to the material distribution and geometric distribution of the spent fuel pool, the overall problem is divided into multiple blocks, and the functional of the neutron flux density distribution is written as the sum of the local functionals of each block; Where v is the number of the node; the functional of a single node is: Discretize the functional of each block: In the formula, f i (r), h γ,l (r) represents the fully orthogonal polynomial basis functions defined in the node and on the node surface, respectively, f, h γ This represents the vector of the expanded basis functions inside the node and on the node surface. s g 、j γ,g are the vectors representing the neutron flux density, the neutron source term and the neutron flux density expansion moment on the nodal boundary surface respectively.

5. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 1 is characterized in that: Step S3 discretizes the functional of each node block and uses piecewise constant function to expand the small group constant to explicitly express the complex inhomogeneity of the material; in, They represent the expansion moment and vector of the minority group constant respectively. In this way, the complex heterogeneity of the material can be explicitly expressed in the discrete form of the functional.

6. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 1 is characterized in that: According to the variational principle, the response relationship between the expanded moment vectors of each variable is obtained: In the formula, Ψ g The vector representing the neutron flux density expansion moment on the node surface. The matrices A and M are determined by the geometry, material and expansion basis functions of the node. Define the neutron flux density expansion moment on the node surface: Substituting into the above formula, we get the matrix equations representing the neutron balance relationship inside the node and the neutron continuity relationship on the node surface: Among them, the response matrices H, C, B, and R of the node are only related to the geometry and material properties; the matrix equations contain the neutron balance relationship inside the node and the neutron flow continuity relationship on the node surface respectively.

7. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 1 is characterized in that: Orthogonal polynomials are used as basis functions for expansion and discretization to obtain the matrix equations of the neutron balance relationship inside the node and the neutron flow continuity relationship on the node surface.

8. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 1 is characterized in that: A hybrid form of first-order hyperbolic function and polynomial function is used as the expansion basis function of neutron flux density and neutron source term in the node to reduce the number of expansion basis functions and reduce the consumption of computing resources.

9. The method for rapidly calculating the effective proliferation factor of a spent fuel pool under emergency conditions based on the neutron diffusion variational block method according to claim 1 is characterized in that: Normal fuel assemblies use uniform minority group constants to establish functionals, and damaged or tilted fuel assemblies use piecewise functions to establish functionals according to actual conditions. Combined with the establishment of global functionals, iterative calculations are performed using the iterative process of the traditional variational nodal method.

10. An application, using the method for rapid calculation of effective proliferation factor of spent fuel pool under emergency conditions based on neutron diffusion variational block method according to any one of claims 1 to 9, characterized in that: Calculate the effective multiplication factor of the spent fuel pool under emergency conditions.

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