Three-dimensional neutron diffusion calculation method for pressurized water reactor fuel assembly under bending condition

Abstract

A three-dimensional neutron diffusion calculation method for a pressurized water reactor fuel assembly under bending condition, according to the bending data of the pressurized water reactor fuel assembly, the pellets of each layer in the axial direction of the reactor core are divided into arbitrary quadrilaterals in the radial direction; the coarse mesh finite difference method is used to solve the bulk average neutron flux density of the arbitrary quadrilateral pellet; based on the conformal transformation method, the arbitrary quadrilateral pellet is mapped into a rectangular pellet and the explicit expression form of the mapping function is determined; the pellet expansion method is used to solve the accurate surface neutron flow density of the rectangular pellet and map back to the arbitrary quadrilateral pellet; the iteration process of the coarse mesh finite difference method of the arbitrary quadrilateral pellet and the expansion method of the rectangular pellet after conformal transformation is constructed; the three-dimensional neutron diffusion calculation under the bending condition of the pressurized water reactor fuel assembly is completed by iteration convergence. The present application is suitable for the neutron diffusion calculation of the commercial pressurized water reactor core, and can provide accurate three-dimensional neutron diffusion calculation results under the bending condition of the pressurized water reactor fuel assembly.

Description

Technical Field

[0001] This invention relates to the field of numerical calculation in nuclear reactor physics, specifically to a three-dimensional neutron diffusion calculation method under bending conditions of pressurized water reactor fuel assemblies. Background Technology

[0002] Fuel assemblies are the core components of a reactor, where nuclear fission reactions occur within the core, releasing nuclear energy. During reactor operation, fuel assemblies deviate from their normal vertical position and bend due to irradiation growth, hydraulic forces, and other factors, resulting in various deformed shapes. Excessive bending can affect the safe operation of the reactor, leading to problems such as difficulties in loading and unloading fuel, excessively high local power, difficulty in inserting control rods, and quadrant power tilting. Therefore, accurately simulating fuel assembly bending is crucial for the numerical simulation and safe operation of pressurized water reactors.

[0003] In commercial pressurized water reactors, a two-step process is typically used for reactor physics numerical calculations to perform functions such as fuel management and nuclear design. In this two-step process, the core calculations require three-dimensional neutron diffusion calculations across all core segments. The core segments are typically neatly arranged with a continuous mesh, each radially representing a standard square the size of a fuel assembly or a quarter of a fuel assembly, and the segment interfaces are continuously intersecting. When fuel assemblies in the reactor core bend, it is difficult to divide all segments into standard square segments. If the segments are re-divided into rectangular segments based on changes in the water gap, the segment sizes vary at different locations, and intersecting segment grids occur. The continuity conditions for neutron flux density and surface neutron flux density between segments no longer apply, and the segment coupling conditions cannot be determined. If the segments are divided into arbitrary quadrilateral segments based on changes in the water gap, the volume average neutron flux density and surface neutron flux density can be obtained using the coarse-grid finite difference method. However, the difference approximation of the neutron flux density leads to a large error, requiring the segment method to obtain the accurate surface neutron flux density and perform iterative solutions. However, deriving and solving the transverse integral equations in the segment method for arbitrary quadrilateral segments is quite difficult. Therefore, existing methods cannot complete accurate three-dimensional neutron diffusion calculations under pressurized water reactor fuel assembly bending conditions. Summary of the Invention

[0004] To overcome the problems existing in the prior art, the present invention aims to provide a three-dimensional neutron diffusion calculation method under the bending condition of pressurized water reactor fuel assemblies. Based on the bending data of the pressurized water reactor fuel assembly, the segments are divided into arbitrary quadrilaterals in the radial direction. The volume average neutron flux density of the arbitrary quadrilateral segments is obtained by the coarse mesh finite difference method. Based on the conformal transformation method, the arbitrary quadrilateral segments are mapped to the rectangular segments corresponding to the boundaries. Based on the segment expansion method, the accurate surface neutron flux density of the rectangular segments is solved and mapped back to the arbitrary quadrilateral segments. A nonlinear iterative process is constructed to solve the arbitrary quadrilateral segment coarse mesh finite difference method and the rectangular segment segment segment expansion method. The three-dimensional neutron diffusion calculation is completed by iterative convergence.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A three-dimensional neutron diffusion calculation method for pressurized water reactor fuel assembly under bending conditions includes the following steps:

[0007] Step 1: Based on the pressurized water reactor fuel assembly bending data, divide the axial layers of the core into arbitrary quadrilaterals in the radial direction, with the interfaces between adjacent arbitrary quadrilaterals continuously intersecting.

[0008] Step 2: Solve for the volume-average neutron flux density of arbitrary quadrilateral nodes using the coarse mesh finite difference method;

[0009] Step 3: Based on the conformal mapping method, map any quadrilateral block to the rectangular block corresponding to the boundary and determine the explicit expression of the mapping function;

[0010] Step 4: Use the nodal expansion method to solve for the accurate surface neutron flux density of the rectangular nodal, and map it back to the arbitrary quadrilateral nodal based on the mapping function obtained in Step 3;

[0011] Step 5: Construct the iterative process for solving the problem using the coarse mesh finite difference method for arbitrary quadrilateral nodal blocks and the rectangular nodal block expansion method after conformal transformation;

[0012] Step 6: Obtain the accurate volume-average neutron flux density and effective multiplication coefficient for each arbitrary quadrilateral segment, and complete the three-dimensional neutron diffusion calculation under the bending condition of the pressurized water reactor fuel assembly.

[0013] Step 2 is as follows:

[0014] 1) Constructing multi-group steady-state neutron diffusion equations for arbitrary quadrilateral nodules

[0015] For any quadrilateral block k divided in step 1, construct the multi-group steady-state neutron diffusion equation as shown in formula (1);

[0016]

[0017] In the formula:

[0018] —Diffusion coefficient of the g-th energy group of any quadrilateral nodal k —— Neutron flux density at position r of the g-th energy group of any quadrilateral nodal k —The expulsion section of the g-th energy group of any quadrilateral nodal k

[0019] G – Total energy group number

[0020] —The scattering cross section from the h-th energy group to the g-th energy group of any quadrilateral nodal k — Neutron flux density υ at position r of energy group h in any quadrilateral block k — Average number of neutrons produced in each fission

[0021] χ g —Fission energy spectrum of the g-th energy group

[0022] k eff —Effective proliferation coefficient

[0023] —Fission cross section of the h-th energy group of any quadrilateral nodal k

[0024] 2) Construct neutron equilibrium equations for arbitrary quadrilateral nodules

[0025] Based on the multi-group steady-state neutron diffusion equation, the neutron equilibrium equation for any quadrilateral block k is obtained by integration over the entire volume of the arbitrary quadrilateral block, as shown in Equation (2).

[0026]

[0027] In the formula:

[0028] —The area of ​​any quadrilateral block k on the interface in the positive direction of i —The area of ​​any quadrilateral segment k on the interface in the negative direction of i.

[0029] V k —Volume of any quadrilateral block k —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k in the negative direction —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k in the positive direction —Volume-average neutron flux density of the g-th energy group of any quadrilateral nodal k —Volume-average neutron flux density of the h-th energy group of any quadrilateral nodal k

[0030] 3) The volume-average neutron flux density is solved by constructing equations using the coarse-mesh finite difference method.

[0031] For any quadrilateral block k, based on the neutron balance equation and combined with the conditions of continuous neutron flux density and continuous surface neutron flux density between quadrilateral blocks, the coarse mesh finite difference method is used to construct equations and solve them to obtain the volume average neutron flux density, fission source term and effective multiplication coefficient of each quadrilateral block; the continuous surface neutron flux density and continuous surface neutron flux density of adjacent faces of quadrilateral blocks k and k+1 are shown in Equation (3) and Equation (4) respectively.

[0032]

[0033]

[0034] In the formula:

[0035] —Discontinuity factor of the interface in the positive direction of the g-th energy group i of any quadrilateral block k —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k. —Discontinuity factor of the interface in the negative direction of the g-th energy group i of any quadrilateral block k+1 —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral node k+1 in the negative direction —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral node k+1 in the negative direction

[0036] For any quadrilateral node k, the coarse mesh finite difference equation is constructed as shown in formula (5);

[0037]

[0038] In the formula:

[0039] — Coarse-net finite-difference coupling factor in the positive direction of the g-th energy group i of any quadrilateral node k —The coupling correction factor in the positive direction of the g-th energy group i of any quadrilateral nodal k, initially set to 0. — Coarse-net finite-difference coupling factor in the negative direction of the g-th energy group of any quadrilateral node k. —The coupling correction factor in the negative direction of the g-th energy group i of any quadrilateral nodal k, initially set to 0. —Discontinuity factor of the interface in the negative direction of the g-th energy group i of any quadrilateral block k —Discontinuity factor of the interface in the positive direction of the g-th energy group i of any quadrilateral segment k-1 —Volume-average neutron flux density of the g-th energy group of any quadrilateral node k+1 —Volume-average neutron flux density of the g-th energy group of any quadrilateral nodal k-1 —The fission source term of the g-th energy group of any quadrilateral block k.

[0040] Step 3 is as follows:

[0041] 1) The Schwarz-Christoffel transformation is used to map the upper half-plane s to the interior of any quadrilateral block of the complex plane t=x+iy. The transformation function is shown in formula (6).

[0042]

[0043] In the formula:

[0044] C1, C2 — Complex constants of the Schwarz-Schristoffel transform

[0045] s j —The point λ on the real axis of the upper half-plane s, corresponding to the vertex j of any quadrilateral segment in the complex plane t. j —The ratio of the interior angle of vertex j of any quadrilateral segment on the complex plane t to π.

[0046] 2) Construct an explicit function to obtain the aspect ratio of the rectangular block corresponding to the conformal transformation of any quadrilateral block. The aspect ratio of the rectangular block is shown in formula (7).

[0047]

[0048] In the formula:

[0049] aspectratio — Aspect ratio of rectangular segment

[0050] 3) Assuming the side length of the standard square block is L, construct rectangular blocks with length aspectratio*L and width L, and rectangular blocks with length L and width L / aspectratio respectively. Use the Schwarz-Christoffel transformation to obtain the transformation function of the upper half plane s to the inside of the rectangular block of the complex plane w=u+iv, as shown in formula (8).

[0051]

[0052] In the formula:

[0053] C3, C4 — Complex constants of the Schwarz-Schristoffel transform

[0054] 4) Based on formulas (6) and (8), obtain the mapping function from any quadrilateral block to a rectangular block, as shown in formula (9);

[0055]

[0056] In the formula:

[0057] g(u,v) is a mapping function from any quadrilateral block to a rectangular block.

[0058] Step 4 is as follows:

[0059] 1) Constructing the multi-group steady-state neutron diffusion equation for rectangular nodules

[0060] For the rectangular block k after conformal transformation, the multi-group steady-state neutron diffusion equation is derived, as shown in Equation (10);

[0061]

[0062] In the formula:

[0063] — Neutron flux density of the g-th energy group at (u,v,z) for rectangular nodal k

[0064] 2) The exact surface neutron flux density of rectangular nodules is solved using the nodal expansion method.

[0065] Based on the multi-group steady-state neutron diffusion equation of rectangular nodal, the transverse integral equations in each direction are obtained. The accurate surface neutron flux density of the rectangular nodal is obtained by solving the average neutron flux density of the arbitrary quadrilateral nodal through the nodal expansion method and step 2. The equation is then mapped back to the arbitrary quadrilateral nodal based on the conformal transformation function in step 3. The transverse integral equation of the rectangular nodal k is shown in formula (11).

[0066]

[0067] In the formula:

[0068] —Partial neutron flux density of the g-th energy group of rectangular nodal k after transverse integration from the interface in the negative direction i to the interface in the positive direction i.

[0069] —The square of the mapping function g 2 (u,v) is the average value after integration from the negative i-direction to the positive i-direction. —Partial neutron flux density of the h-th energy group of rectangular nodal k after transverse integration from the interface in the negative direction i to the interface in the positive direction i.

[0070] —The leakage term of the transverse integral after transverse integration of the g-th energy group of the rectangular nodal block k from the interface in the negative direction i to the interface in the positive direction i.

[0071] Step 5 is as follows:

[0072] 1) Solving the finite difference equation of an arbitrary quadrilateral nodal coarse mesh based on accurate surface neutron flux density updates

[0073] Based on the accurate surface neutron flux density of the arbitrary quadrilateral block obtained in step 4, the coupling correction factor in the coarse mesh finite difference equation of the arbitrary quadrilateral block is updated and the solution of the coarse mesh finite difference equation of the arbitrary quadrilateral block is updated; the update of the coupling correction factor of the arbitrary quadrilateral block k is shown in formula (12) and formula (13).

[0074]

[0075] In the formula:

[0076] —The precise surface neutron flux density of the interface in the positive direction of the g-th energy group i of any quadrilateral nodal k obtained by the nodal expansion method.

[0077] — Coarse-net finite-difference coupling factor in the positive direction of the g-th energy group of arbitrary quadrilateral nodal k-1 —The precise surface neutron flux density at the interface of the g-th energy group i in the positive direction of an arbitrary quadrilateral nodal k-1 obtained by the nodal expansion method.

[0078] 2) Determine whether the effective multiplication coefficient and fission source term before and after the update meet the convergence conditions.

[0079] Calculate whether the effective multiplication coefficient and the relative deviation of the fission source terms of each arbitrary quadrilateral block before and after updating the coarse mesh finite difference equation of the arbitrary quadrilateral block meet the convergence condition. If the convergence condition is not met, use the updated average neutron flux density of the arbitrary quadrilateral block to iteratively execute steps 4 and 5 until the convergence condition is met; if the convergence condition is met, execute step 6. The convergence condition is that the effective multiplication coefficient and the relative deviation of the fission source terms of each arbitrary quadrilateral block before and after updating the coarse mesh finite difference equation of the arbitrary quadrilateral block are less than the convergence limit, as shown in formula (14) and formula (15), respectively.

[0080]

[0081]

[0082] In the formula:

[0083] —Updated effective multiplication coefficient

[0084] —Effective proliferation coefficient before update

[0085] ε1 — Convergence limit of the effective multiplication coefficient

[0086] —The updated fission source term of the g-th energy group for any quadrilateral nodal k —The fission source term of the g-th energy group of any quadrilateral block k before the update

[0087] ε2 — Convergence limit of the fission source term

[0088] Compared with the prior art, the present invention has the following advantages:

[0089] 1. Based on actual fuel assembly bending data, each layer of the core is divided into arbitrary quadrilaterals in the radial direction. The interface between any two segments is continuously intersecting to avoid grid intersection in the segmented segments. The conditions for continuous surface neutron flux density and continuous surface neutron flow density between arbitrary quadrilateral segments are still applicable.

[0090] 2. For any quadrilateral nodule, a conformal transformation method is used to map it to a rectangular nodule. Then, the accurate surface neutron flux density is solved by the nodule expansion method and mapped back to the arbitrary quadrilateral nodule. This avoids directly deriving and solving the transverse integral equation in the nodule expansion method for the arbitrary quadrilateral nodule, thus achieving accurate three-dimensional neutron diffusion calculation. Attached Figure Description

[0091] Figure 1 A flowchart for three-dimensional neutron diffusion calculation under bending conditions of pressurized water reactor fuel assemblies;

[0092] Figure 2 A radial diagram illustrating the bending and arbitrary quadrilateral segmentation of a 3×3 fuel assembly;

[0093] Figure 3 A radial schematic diagram showing the continuity of surface neutron flux density and surface neutron current density on adjacent faces of arbitrary quadrilateral nodes k and k+1.

[0094] Figure 4 A radial diagram showing how any quadrilateral segment on the complex plane t is mapped to a rectangular segment on the complex plane w via the upper half-plane s;

[0095] Figure 5 A radial schematic diagram to verify the conventional core model;

[0096] Figure 6 A radial schematic diagram to verify the core bending model;

[0097] Figure 7 To realize the power distribution change calculated by the pressurized water reactor core calculation program SPARK before this method (to verify the core bending model compared to the conventional core model);

[0098] Figure 8 To realize the power distribution change calculated by the pressurized water reactor core calculation program SPARK after this method (to verify the core bending model compared with the conventional core model);

[0099] Figure 9 The power distribution variation calculated for the Monte Carlo program (validating the core bending model compared to the validating core conventional model). Detailed Implementation

[0100] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0101] This invention discloses a three-dimensional neutron diffusion calculation method under the bending condition of pressurized water reactor fuel assemblies. Based on the bending data of the pressurized water reactor fuel assemblies, the core segments are radially divided into arbitrary quadrilaterals. The volume-average neutron flux density of the arbitrary quadrilateral segments is obtained using a coarse-mesh finite difference method. Based on a conformal transformation method, the arbitrary quadrilateral segments are mapped to corresponding rectangular segments. The accurate surface neutron flux density of the rectangular segments is solved using a segment expansion method and mapped back to the arbitrary quadrilateral segments. An iterative process is constructed for solving the arbitrary quadrilateral segment coarse-mesh finite difference method and the rectangular segment expansion method. The iterative convergence completes the three-dimensional neutron diffusion calculation. Specific steps are as follows: Figure 1 As shown;

[0102] Step 1: Based on the pressurized water reactor fuel assembly bending data, the axially arranged segments of the core, calculated using three-dimensional neutron diffusion, are radially divided into arbitrary quadrilaterals, such as... Figure 2 The diagram shows a radial diagram of the bending of a 3×3 fuel assembly and the division of arbitrary quadrilateral segments. The standard square segment in the diagram is radially taken as the size of one fuel assembly. When fuel assembly 1 bends upward and fuel assembly 2 bends to the right, the center point of the vertices of the adjacent bent fuel assemblies is selected as the vertex of the arbitrary quadrilateral segment, and the segments are re-divided into arbitrary quadrilaterals, so that the interfaces between adjacent arbitrary quadrilateral segments intersect continuously.

[0103] Step 2: Solve for the volume-average neutron flux density of arbitrary quadrilateral nodes using the coarse mesh finite difference method; this step is specifically implemented in the following 3 steps:

[0104] 1) Constructing multi-group steady-state neutron diffusion equations for arbitrary quadrilateral nodules

[0105] For any quadrilateral block k divided in step 1, construct the multi-group steady-state neutron diffusion equation as shown in formula (1);

[0106]

[0107] In the formula:

[0108] —Diffusion coefficient of the g-th energy group of any quadrilateral nodal k —— Neutron flux density at position r of the g-th energy group of any quadrilateral nodal k —The expulsion section of the g-th energy group of any quadrilateral nodal k

[0109] G – Total energy group number

[0110] —The scattering cross section from the h-th energy group to the g-th energy group of any quadrilateral nodal k — Neutron flux density υ at position r of energy group h in any quadrilateral block k — Average number of neutrons produced in each fission

[0111] χ g —Fission energy spectrum of the g-th energy group

[0112] k eff —Effective proliferation coefficient

[0113] —Fission cross section of the h-th energy group of any quadrilateral nodal k

[0114] 2) Construct neutron equilibrium equations for arbitrary quadrilateral nodules

[0115] Based on the multi-group steady-state neutron diffusion equation, the neutron equilibrium equation for any quadrilateral block k is obtained by integration over the entire volume of the arbitrary quadrilateral block, as shown in Equation (2).

[0116]

[0117] In the formula:

[0118] —The area of ​​any quadrilateral block k on the interface in the positive direction of i —The area of ​​any quadrilateral segment k on the interface in the negative direction of i.

[0119] V k —Volume of any quadrilateral block k —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k in the negative direction —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k in the positive direction —Volume-average neutron flux density of the g-th energy group of any quadrilateral nodal k —Volume-average neutron flux density of the h-th energy group of any quadrilateral nodal k

[0120] 3) The volume-average neutron flux density is solved by constructing equations using the coarse-mesh finite difference method.

[0121] For any quadrilateral node k, based on the neutron balance equation and combining the conditions of continuous neutron flux density and continuous surface neutron flux density between arbitrary quadrilateral nodes, a coarse mesh finite difference method is used to construct and solve equations to obtain the volume-average neutron flux density, fission source term, and effective multiplication coefficient for each arbitrary quadrilateral node; such as Figure 3As shown, this is a radial schematic diagram of the continuity of surface neutron flux density and surface neutron flow on adjacent faces of arbitrary quadrilateral blocks k and k+1. In the figure, the surface neutron flux density and surface neutron flow density of arbitrary quadrilateral block k in the positive direction of i are continuous with the surface neutron flux density and surface neutron flow density of arbitrary quadrilateral block k+1 in the negative direction of i. The continuity of surface neutron flux density and surface neutron flow density are shown in Equation (3) and Equation (4), respectively.

[0122]

[0123]

[0124] In the formula:

[0125] —Discontinuity factor of the interface in the positive direction of the g-th energy group i of any quadrilateral block k —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral block k. —Discontinuity factor of the interface in the negative direction of the g-th energy group i of any quadrilateral block k+1 —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral node k+1 in the negative direction —Surface neutron flux density at the interface of the g-th energy group i in any quadrilateral node k+1 in the negative direction

[0126] For any quadrilateral node k, the coarse mesh finite difference equation is constructed as shown in formula (5);

[0127]

[0128] In the formula:

[0129] — Coarse-net finite-difference coupling factor in the positive direction of the g-th energy group i of any quadrilateral node k —The coupling correction factor in the positive direction of the g-th energy group i of any quadrilateral nodal k, initially set to 0. — Coarse-net finite-difference coupling factor in the negative direction of the g-th energy group of any quadrilateral node k. —The coupling correction factor in the negative direction of the g-th energy group i of any quadrilateral nodal k, initially set to 0. —Discontinuity factor of the interface in the negative direction of the g-th energy group i of any quadrilateral block k —Discontinuity factor of the interface in the positive direction of the g-th energy group i of any quadrilateral segment k-1 —Volume-average neutron flux density of the g-th energy group of any quadrilateral node k+1 —Volume-average neutron flux density of the g-th energy group of any quadrilateral nodal k-1 —The fission source term of the g-th energy group of any quadrilateral block k

[0130] Step 3: Based on the conformal mapping method, map any quadrilateral block to the corresponding rectangular block at the boundary and determine the explicit expression of the mapping function; such as Figure 4 As shown, the arbitrary quadrilateral blocks on the upper half-plane s and the rectangular blocks on the complex plane t are transformed respectively, and the explicit expression of the mapping function is determined according to the transformation function; this step is specifically implemented in the following 3 steps;

[0131] 1) The Schwarz-Christoffel transformation is used to map the upper half-plane s to the interior of any quadrilateral block of the complex plane t=x+iy. The transformation function is shown in formula (6).

[0132]

[0133] In the formula:

[0134] C1, C2 — Complex constants of the Schwarz-Schristoffel transform

[0135] s j —The point λ on the real axis of the upper half-plane s, corresponding to the vertex j of any quadrilateral segment in the complex plane t. j —The ratio of the interior angle of vertex j of any quadrilateral segment on the complex plane t to π.

[0136] 2) Construct an explicit function to obtain the aspect ratio of the rectangular block corresponding to the conformal transformation of any quadrilateral block. The aspect ratio of the rectangular block is shown in formula (7).

[0137]

[0138] In the formula:

[0139] aspectratio — Aspect ratio of rectangular segment

[0140] 3) Assuming the side length of the standard square block is L, construct rectangular blocks with length aspectratio*L and width L, and rectangular blocks with length L and width L / aspectratio respectively. Use the Schwarz-Christoffel transformation to obtain the transformation function of the upper half plane s to the inside of the rectangular block of the complex plane w=u+iv, as shown in formula (8).

[0141]

[0142] In the formula:

[0143] C3, C4 — Complex constants of the Schwarz-Schristoffel transform

[0144] 4) Based on formulas (6) and (8), obtain the mapping function from any quadrilateral block to a rectangular block, as shown in formula (9);

[0145]

[0146] In the formula:

[0147] g(u,v) — A mapping function from any quadrilateral segment to a rectangular segment.

[0148] Step 4: Since the coarse mesh finite difference method uses a difference approximation for the neutron flux density, the error in the surface neutron flux density is relatively large. Therefore, the nodal expansion method is used to solve the accurate surface neutron flux density of the rectangular nodal, and the mapping function obtained in step 3 is used to map back to the arbitrary quadrilateral nodal. This step is specifically implemented in the following two steps.

[0149] 1) Constructing the multi-group steady-state neutron diffusion equation for rectangular nodules

[0150] For the rectangular block k after conformal transformation, the multi-group steady-state neutron diffusion equation is derived, as shown in Equation (10);

[0151]

[0152] In the formula:

[0153] — Neutron flux density of the g-th energy group at (u,v,z) for rectangular nodal k

[0154] 2) The exact surface neutron flux density of rectangular nodules is solved using the nodal expansion method.

[0155] Based on the multi-group steady-state neutron diffusion equation of rectangular nodal, the transverse integral equations in each direction are obtained. The accurate surface neutron flux density of the rectangular nodal is obtained by solving the average neutron flux density of the arbitrary quadrilateral nodal through the nodal expansion method and step 2. The equation is then mapped back to the arbitrary quadrilateral nodal based on the conformal transformation function in step 3. The transverse integral equation of the rectangular nodal k is shown in formula (11).

[0156]

[0157] In the formula:

[0158] —Partial neutron flux density of the g-th energy group of rectangular nodal k after transverse integration from the interface in the negative direction i to the interface in the positive direction i.

[0159] —The square of the mapping function g 2 (u,v) is the average value after integration from the negative i-direction to the positive i-direction. —Partial neutron flux density of the h-th energy group of rectangular nodal k after transverse integration from the interface in the negative direction i to the interface in the positive direction i.

[0160] —The transverse integral leakage term after transverse integration of the g-th energy group of the rectangular nodal k from the interface in the negative direction i to the interface in the positive direction i.

[0161] Step 5: Construct the iterative process for solving the problem using the coarse mesh finite difference method for arbitrary quadrilateral nodal blocks and the rectangular nodal expansion method after conformal transformation; this step is specifically implemented in the following two steps;

[0162] 1) Solving the finite difference equation of an arbitrary quadrilateral nodal coarse mesh based on accurate surface neutron flux density updates

[0163] Based on the accurate surface neutron flux density of the arbitrary quadrilateral block obtained in step 4, the coupling correction factor in the coarse mesh finite difference equation of the arbitrary quadrilateral block is updated and the solution of the coarse mesh finite difference equation of the arbitrary quadrilateral block is updated; the update of the coupling correction factor of the arbitrary quadrilateral block k is shown in formula (12) and formula (13).

[0164]

[0165]

[0166] In the formula:

[0167] —The precise surface neutron flux density of the interface in the positive direction of the g-th energy group i of any quadrilateral nodal k obtained by the nodal expansion method.

[0168] — Coarse-net finite-difference coupling factor in the positive direction of the g-th energy group of arbitrary quadrilateral nodal k-1 —The precise surface neutron flux density at the interface of the g-th energy group i in the positive direction of an arbitrary quadrilateral nodal k-1 obtained by the nodal expansion method.

[0169] 2) Determine whether the effective multiplication coefficient and fission source term before and after the update meet the convergence conditions.

[0170] The effective multiplication coefficients before and after updating the coarse mesh finite difference equation of arbitrary quadrilateral blocks and the relative deviations of the fission source terms of each arbitrary quadrilateral block are calculated to see if they meet the convergence conditions, as shown in Equation (14) and Equation (15), respectively.

[0171]

[0172]

[0173] In the formula:

[0174] —Updated effective multiplication coefficient —Effective proliferation coefficient before update

[0175] ε1 — Convergence limit of the effective multiplication coefficient

[0176] —The updated fission source term of the g-th energy group for any quadrilateral nodal k —The fission source term of the g-th energy group of any quadrilateral block k before the update

[0177] ε2 — Convergence limit of the fission source term

[0178] If the convergence condition is not met, then use the updated average neutron flux density of the arbitrary quadrilateral block to iteratively execute steps 4 and 5 until the convergence condition is met; if the convergence condition is met, then execute step 6.

[0179] Step 6: Obtain the accurate volume-average neutron flux density and effective multiplication coefficient for each arbitrary quadrilateral segment, and complete the three-dimensional neutron diffusion calculation under the bending condition of the pressurized water reactor fuel assembly.

[0180] After implementing the above methods and steps in the pressurized water reactor core calculation program SPARK, a verification core conventional model and a bending model were constructed for calculation, such as... Figure 5 The diagram shown is a radial schematic for verifying the conventional core model; as shown... Figure 6 The diagram shown is a radial schematic for verifying the core bending model, where the surrounding components of the central core assembly bend outwards with the same water gap width. The above model was simulated using the pressurized water reactor core calculation program SPARK before and after implementing the method of this invention, and the k-value was analyzed. eff The changes in the bending model and power distribution are compared with the Monte Carlo method as a reference solution. Table 1 shows the differences between the bending model calculated by different methods and the conventional model k. eff Contrast of changes; such as Figure 7 As shown, this illustrates the power distribution change calculated by the SPARK core calculation program before implementing the method of this invention; Figure 8 The image shows the power distribution change calculated by the SPARK core calculation program after implementing the method of this invention; as shown... Figure 9 The figure shows the power distribution variation calculated by the Monte Carlo program.

[0181] Table 1

[0182] SPARK (before improvement) -15 SPARK (Improved) -44 Monte Carlo Program -73

[0183] Numerical results show that, after adopting the method of this invention, the SAPRK core calculation program for pressurized water reactors calculates and verifies the core bending model compared to the conventional core verification model. effThe power distribution variation is closer to the Monte Carlo program reference solution. Therefore, the method of this invention is applicable to commercial pressurized water reactor core calculations, and can provide accurate three-dimensional neutron diffusion calculation results under pressurized water reactor fuel assembly bending conditions, thus possessing industrial application value.

Claims

1. A three-dimensional neutron diffusion calculation method under the bending condition of a pressurized water reactor fuel assembly, characterized in that: The steps include the following: Step 1: Based on the pressurized water reactor fuel assembly bending data, divide the axial layers of the core into arbitrary quadrilaterals in the radial direction, with the interfaces between adjacent arbitrary quadrilaterals continuously intersecting. Step 2: Solve for the volume-average neutron flux density of arbitrary quadrilateral nodes using the coarse mesh finite difference method; Step 3: Based on the conformal mapping method, map any quadrilateral block to the corresponding rectangular block at the boundary and determine the explicit expression of the mapping function; specifically as follows: 1) Use the Schwarz-Christoffel transform to transform the upper half-plane s Mapping to the complex plane t The transformation function inside any quadrilateral block is shown in formula (6); Official (6) In the formula: , —— Complex constants of the Schwarz-Schristoffel transform — Upper half-plane s Corresponding complex plane on the real axis t Vertex of any quadrilateral block j point —— Complex plane t Vertex of any quadrilateral block j interior angles and π ratio 2) Construct an explicit function to obtain the aspect ratio of the rectangular block corresponding to the conformal transformation of any quadrilateral block. The aspect ratio of the rectangular block is shown in formula (7). Official (7) In the formula: aspectratio — Aspect ratio of rectangular segments 3) Assume the side length of the standard square section is... L Construct lengths of respectively aspectratio * L Width L rectangular blocks and lengths of L Width L / aspectratio The upper half-plane is obtained from the rectangular sections using the Schwarz-Christoffel transform. s Mapping to the complex plane w The transformation function inside the rectangular block is shown in formula (8); Official (8) In the formula: , —— Complex constants of the Schwarz-Schristoffel transform 4) Based on formulas (6) and (8), obtain the mapping function from any quadrilateral block to a rectangular block, as shown in formula (9); Official (9) In the formula: g ( u , v — A mapping function from any quadrilateral block to a rectangular block; Step 4: Use the nodal expansion method to solve for the accurate surface neutron flux density of the rectangular nodal, and map it back to the arbitrary quadrilateral nodal based on the mapping function obtained in Step 3; Step 5: Construct the iterative process for solving the problem using the coarse mesh finite difference method for arbitrary quadrilateral nodal blocks and the rectangular nodal block expansion method after conformal transformation; Step 6: Obtain the accurate volume-average neutron flux density and effective multiplication coefficient for each arbitrary quadrilateral segment, and complete the three-dimensional neutron diffusion calculation under the bending condition of the pressurized water reactor fuel assembly.

2. The method for calculating three-dimensional neutron diffusion under bending conditions of pressurized water reactor fuel assemblies according to claim 1, characterized in that: Step 2 is as follows: 1) Constructing multi-group steady-state neutron diffusion equations for arbitrary quadrilateral nodules For any quadrilateral segment divided in step 1 k The multi-group steady-state neutron diffusion equation is constructed as shown in equation (1); Official (1) In the formula: — Arbitrary quadrilateral blocks k No. g diffusion coefficient of energy group — Arbitrary quadrilateral blocks k No. g Energy Group r Neutron flux density at location — Arbitrary quadrilateral blocks k No. g energy group moveout cross section G — Total number of energy groups —Arbitrary quadrilateral block k No. h The group can reach the first g Scattering cross section of energy group — Arbitrary quadrilateral blocks k No. h Energy Group r Neutron flux density at location — The average number of neutrons produced per fission cycle ——No. g fission energy spectrum of energy group — Effective multiplication coefficient — Arbitrary quadrilateral blocks k No. h fission cross section of energy group 2) Constructing neutron equilibrium equations for arbitrary quadrilateral nodules Based on the multi-group steady-state neutron diffusion equation, the arbitrary quadrilateral segment is obtained by integration over the entire volume of the arbitrary quadrilateral segment. k The neutron balance equation is shown in Equation (2); Official (2) In the formula: — Arbitrary quadrilateral blocks k exist i Area of ​​the positive direction interface — Arbitrary quadrilateral blocks k exist i Area of ​​the negative direction interface V k — Arbitrary quadrilateral blocks k volume — Arbitrary quadrilateral blocks k The g-th energy group i Surface neutron flux density at the negative direction interface — Arbitrary quadrilateral blocks k The g-th energy group i Surface neutron flux density at the positive direction interface — Arbitrary quadrilateral blocks k The volume average neutron flux density of the g-th energy group — Arbitrary quadrilateral blocks k No. h Volume average neutron flux density of the energy group 3) The volume-average neutron flux density is solved by constructing equations using the coarse-mesh finite difference method. For any quadrilateral block k Based on the neutron balance equation and combining the conditions of continuous neutron flux density and continuous surface neutron flux density between arbitrary quadrilateral nodes, the coarse mesh finite difference method is used to construct and solve the equations to obtain the volume-average neutron flux density, fission source term, and effective multiplication coefficient for each arbitrary quadrilateral node; arbitrary quadrilateral nodes k and k The continuity of the surface neutron flux density and the continuity of the surface neutron flux density of the +1 adjacent surfaces are shown in Equation (3) and Equation (4), respectively. Official (3) Official (4) In the formula: — Arbitrary quadrilateral blocks k No. g Energy Group i Discontinuity factor of the positive direction interface — Arbitrary quadrilateral blocks k No. g Energy Group i Neutron flux density at the positive direction interface — Arbitrary quadrilateral blocks k+ 1st g Energy Group i Discontinuity factor of negative direction interface — Arbitrary quadrilateral blocks k+ 1st g Energy Group i Surface neutron flux density at the negative direction interface — Arbitrary quadrilateral blocks k +1th g Energy Group i Surface neutron flux density at the negative direction interface For any quadrilateral segment k The constructed coarse mesh finite difference equation is shown in formula (5); Official (5) In the formula: — Arbitrary quadrilateral blocks k No. g Energy Group i positive direction coarse mesh finite difference coupling factor — Arbitrary quadrilateral blocks k No. g Energy Group i The positive coupling correction factor is initially set to 0. — Arbitrary quadrilateral blocks k No. g Energy Group i Coarse-net finite-difference coupling factor in the negative direction — Arbitrary quadrilateral blocks k No. g Energy Group i The negative coupling correction factor is initially set to 0. — Arbitrary quadrilateral blocks k No. g Energy Group i Discontinuity factor of negative direction interface — Arbitrary quadrilateral blocks k -1st g Energy Group i Discontinuity factor of the positive direction interface — Arbitrary quadrilateral blocks k +1th g Volume average neutron flux density of the energy group — Arbitrary quadrilateral blocks k -1st g Volume average neutron flux density of the energy group — Arbitrary quadrilateral blocks k No. g The fission source term of the energy group.

3. The method for calculating three-dimensional neutron diffusion under bending conditions of pressurized water reactor fuel assemblies according to claim 2, characterized in that: Step 4 is as follows: 1) Constructing the multi-group steady-state neutron diffusion equation for rectangular nodules For rectangular blocks after conformal transformation k The multi-group steady-state neutron diffusion equation is derived, as shown in equation (10); Official (10) In the formula: — Rectangular section k exist( u , v , z ) g Neutron flux density of energy group 2) The exact surface neutron flux density of rectangular nodules is solved using the nodal expansion method. Based on the multi-group steady-state neutron diffusion equation of rectangular nodes, transverse integral equations in each direction are obtained. The accurate surface neutron flux density of the rectangular nodes is obtained by solving the volume-average neutron flux density of the arbitrary quadrilateral nodes using the node expansion method and step 2, and then mapped back to the arbitrary quadrilateral nodes based on the conformal transformation function in step 3. k The horizontal integral equation is shown in formula (11); Official (11) In the formula: — Rectangular section k No. g Energy group from i Negative direction interface to i Partial neutron flux density after transverse integration of the positive direction interface — The square of the mapping function g 2 ( u , v )from i negative direction to i The average value after integration in the positive direction — Rectangular section k No. h Energy group from i Negative direction interface to i Partial neutron flux density after transverse integration of the positive direction interface — Rectangular section k No. g Energy group from i Negative direction interface to i Leaked terms in the horizontal integral after horizontal integration in the positive direction interface.

4. The method for calculating three-dimensional neutron diffusion under bending conditions of pressurized water reactor fuel assemblies according to claim 2, characterized in that: Step 5 is as follows: 1) Solving the finite difference equation of an arbitrary quadrilateral nodal coarse mesh based on accurate surface neutron flux density updates Based on the accurate surface neutron flux density of the arbitrary quadrilateral nodal obtained in step 4, the coupling correction factor in the coarse mesh finite difference equation of the arbitrary quadrilateral nodal is updated and the solution of the coarse mesh finite difference equation of the arbitrary quadrilateral nodal is updated. Arbitrary quadrilateral block k The updating of the coupling correction factor is shown in Equations (12) and (13); Official (12) Official (13) In the formula: — Arbitrary quadrilateral nodes obtained by the node unfolding method k No. g Energy Group i Precise surface neutron flux density at the positive direction interface — Arbitrary quadrilateral blocks k -1st g Energy Group i positive direction coarse mesh finite difference coupling factor — Arbitrary quadrilateral nodes obtained by the node unfolding method k -1st g Energy Group i Precise surface neutron flux density at the positive direction interface 2) Determine whether the effective multiplication coefficient and fission source term before and after the update satisfy the convergence condition. Calculate whether the effective multiplication coefficients before and after updating the coarse mesh finite difference equation of the arbitrary quadrilateral block and the relative deviations of the fission source terms of each arbitrary quadrilateral block satisfy the convergence condition. If the convergence condition is not satisfied, use the updated average neutron flux density of the arbitrary quadrilateral block to iteratively execute steps 4 and 5 until the convergence condition is satisfied. If the convergence condition is satisfied, execute step 6.

5. The method for calculating three-dimensional neutron diffusion under bending conditions of pressurized water reactor fuel assemblies according to claim 4, characterized in that: The convergence condition is that the effective multiplication coefficients before and after updating the coarse mesh finite difference equation of the arbitrary quadrilateral block and the relative deviations of the fission source terms of each arbitrary quadrilateral block are less than the convergence limit, as shown in formula (14) and formula (15), respectively. Official (14) Official (15) In the formula: — Updated effective multiplication coefficient — Effective proliferation coefficient before update — Convergence limit of the effective multiplication coefficient — Updated arbitrary quadrilateral blocks k No. g fission source term of energy group — Arbitrary quadrilateral blocks before the update k No. g fission source term of energy group — Convergence limit of the fission source term.

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