Method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on the thermal conductivity

Abstract

The application discloses a method for determining the influence of the irradiation densification degree of ceramic nuclear fuel on the heat conduction performance, collecting physical property parameters in the irradiation process of the ceramic nuclear fuel, and constructing a free energy equation of pore and grain evolution in the irradiation densification process; the initial pore distribution is obtained by simulating the sintering preparation process of the ceramic nuclear fuel, and the initial pore distribution is taken as the initial condition of the irradiation densification simulation; the evolution law of point defects in the irradiation densification process is obtained through the rate theory; the phase field evolution equation is constructed and solved on the basis of introducing the irradiation condition and the polycrystalline structure; the numerical solution obtained after solving the evolution equation is subjected to visual processing, the pore morphology and the evolution law in the irradiation densification process of the ceramic nuclear fuel are obtained, the number and size of the pores are counted, the porosity change and the densification law are obtained, the heat conduction law in the densification process of the ceramic nuclear fuel is obtained through the statistical pore distribution, and the service performance of the ceramic nuclear fuel is predicted.

Description

Technical Field

[0001] This invention relates to the field of nuclear fuel technology, and more specifically to a method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on its thermal conductivity. Background Technology

[0002] In the field of nuclear energy, irradiation densification is an important research topic, having a crucial impact on nuclear fuel performance and reactor safety. Irradiation densification refers to a kinetic process initiated by the radiation of high-energy neutrons in the reactor during the initial stages of nuclear fuel service. During this process, the initial pores in sintered uranium dioxide nuclear fuel gradually shrink under the influence of the neutron flux during irradiation, leading to an increase in the density of the fuel elements. Furthermore, as burnup increases, pores that remain after shrinkage become potential traps for gas atoms, and the pore shrinkage caused by irradiation densification is eventually replaced by pore growth caused by irradiation swelling. However, current research still presents many controversies and challenges regarding the generation mechanism of irradiation densification and the relationship between irradiation densification and the transition from irradiation swelling. Research on irradiation densification aims to gain a deeper understanding of the behavior of nuclear fuel in the nuclear reactor environment and to provide a theoretical basis and technical support for improving the efficiency and safety of nuclear reactors. In addition, maintaining an appropriate temperature distribution is crucial for the stable operation of nuclear fuel. Uneven temperature distribution can lead to overheating of some fuel elements, causing damage to the fuel elements or failure of the fuel cladding, and even potentially resulting in a serious nuclear reactor accident. Therefore, the thermal conductivity of nuclear fuel directly affects the safety and stability of a nuclear reactor. Irradiation densification can alter the fuel structure, thereby affecting its thermal conductivity. Therefore, in-depth research into irradiation densification and its impact on thermal conductivity will help to better understand the behavior of nuclear fuel under irradiation conditions and provide more accurate references and guidance for the design and operation of nuclear reactors. Summary of the Invention

[0003] The purpose of this invention is to provide a phase-field simulation method for studying the initial porosity shrinkage of uranium dioxide nuclear fuel after irradiation and its impact on thermal conductivity. Specifically, it is a method for determining the degree of nuclear fuel irradiation densification and its impact on thermal conductivity. This method can dynamically reproduce the evolution process of pores in nuclear fuel under irradiation conditions, providing a predictive method for nuclear fuel irradiation densification and its impact on thermal conductivity.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on its thermal conductivity includes the following steps:

[0006] S1 selects phase field variables to describe the densification process of ceramic nuclear fuel under irradiation; collects physical property parameters of ceramic nuclear fuel, including the formation energy of vacancies, interstitial atoms and gas atoms, the diffusion coefficient and gradient coefficient of ceramic nuclear fuel after irradiation acceleration, and the mobility of pores and grain interfaces.

[0007] S2 obtains the free energy density function of the ceramic nuclear fuel matrix phase based on the material thermodynamics theory, according to the physical property parameters of the ceramic nuclear fuel collected in step S1. By combining the phase field empirical formula, the free energy density function of the porous phase in the ceramic nuclear fuel is obtained. Then, the total free energy equation is obtained by combining the polycrystalline interaction energy and the interface gradient energy.

[0008] S3 establishes a sintering phase field model that considers the rigid motion of ceramic nuclear fuel particles, simulates the physical process of sintering ceramic nuclear fuel particles into nuclear fuel, obtains the pore distribution at the moment when the sintering preparation of ceramic nuclear fuel is completed, and uses this pore distribution as the initial structure for the simulation of the irradiation densification phase field of ceramic nuclear fuel.

[0009] S4 uses the connected region labeling method to statistically analyze the pore radius distribution during the irradiation densification process of ceramic nuclear fuel; then, combined with rate theory, it analyzes the densification mechanism and obtains the time-dependent variation law of point defects in densification under irradiation conditions.

[0010] Based on the total free energy equation obtained in step S2 and the point defect dynamics equation obtained in step S4, and taking into account the generation and annihilation of point defects under irradiation conditions and the absorption of point defects by grain boundaries, S5 constructs an irradiation densification phase field evolution equation that varies with burnup; the semi-implicit Fourier spectroscopy method is used to solve the phase field evolution equation to obtain a numerical solution.

[0011] S6 visualizes the numerical solution obtained in step S5 to obtain the morphology of the initial pore compaction process during the compaction of ceramic nuclear fuel; and statistically analyzes the porosity at different times to obtain the compaction law of ceramic nuclear fuel irradiation.

[0012] Based on the obtained pore distribution and porosity, S7 coupled the phase field-based thermal conductivity calculation formula to obtain the thermal conductivity variation law during the irradiation densification process of ceramic nuclear fuel.

[0013] In one implementation method, S1 specifically includes:

[0014] S1.1 Selecting phase field variables to describe the densification process of ceramic nuclear fuel under irradiation requires selecting phase field variables to represent point defect concentration: vacancy concentration c. v Interstitial atom concentration c i and gas atom concentration c gSimultaneously, a porosity phase sequence parameter η is selected to distinguish between the porosity phase and the matrix phase: η = 1.0 when located in the porosity phase, and η = 0.0 when located in the matrix phase; a set of sequence parameters φ also needs to be selected. i This indicates the grain with index i, and the index parameter indicates whether there is φ in the i-th grain. i =1.0;

[0015] S1.2 Collect the physical properties of nuclear fuel, including the formation energies of vacancies, interstitial atoms and gas atoms, the diffusion coefficient and gradient coefficient of ceramic nuclear fuel, and the mobility of pores and grain interfaces.

[0016] S1.3 Based on the diffusion coefficients of vacancies, interstitial atoms, and gas atoms in S1.2, calculate the mobility considering point defect irradiation-accelerated diffusion. The diffusion coefficient of atoms in the crystal matrix is ​​expressed as:

[0017]

[0018] Where θ is a material-related constant; ν is the vibrational frequency of the atom; G m k is the free energy required for an atom to migrate from its equilibrium position to its nearest position. B Here, c is Boltzmann's constant, T is absolute temperature; v Let be the vacancy concentration; then the vacancy diffusion coefficient after irradiation acceleration is written as:

[0019]

[0020] Where c v e c is the thermal equilibrium concentration of vacancies. v r The vacancy concentration introduced by irradiation; D0 is the pre-factor of the diffusion coefficient; E i To activate energy;

[0021] Vacancy mobility M v The following expression exists:

[0022]

[0023] Where D v To account for the vacancy diffusion coefficient after irradiation acceleration; V m R is the lattice volume; R is the ideal gas constant.

[0024] In one implementation method, S2 specifically includes:

[0025] Based on the thermodynamics of materials, the free energy density function of the ceramic nuclear fuel matrix phase is calculated using the following formula:

[0026]

[0027] Among them, c v m c i m c g m These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the matrix phase, respectively. These represent the formation energies of vacancies, interstitial atoms, and gas atoms, respectively; k B Boltzmann constant; T is absolute temperature;

[0028] The free energy density function of the porous phase of S2.2 ceramic nuclear fuel is as follows:

[0029]

[0030] Among them, c v b c i b c g b These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the porous phase, respectively, where A and B are constants, and f is the concentration of gas atoms in the porous phase. g b (c g b Let f be the gas Gibbs free energy function related to the gas atom concentration, obtained through reference thermophysics. g b (c g b The expression for ) is:

[0031]

[0032] Among them, V a The volume of a single U atom; n Q b is the quantum concentration; b is the ideal gas constant.

[0033] S2.3 Based on the obtained free energy density functions of the matrix phase and the porosity phase, an interpolation function is introduced to obtain the bulk free energy density function:

[0034]

[0035] Where h(η) = η 3 (6η 2 The interpolation function (-15η+10) is constructed. When η = 1.0, it satisfies h(η) = 1.0; and when η = 0.0, it satisfies h(η) = 0.0. The relationship between the total defect concentration and the defect concentration in each phase is as follows:

[0036]

[0037] When ceramic nuclear fuel reaches equilibrium, the chemical potentials corresponding to the matrix phase and the pores are equal, therefore:

[0038]

[0039] The polycrystalline free energy density function is:

[0040]

[0041] Where α, β, γ, and δ are phenomenological parameters; φ i Indicates the grain with serial number i; φ j This indicates the grain with serial number j;

[0042] Combining bulk free energy density, interpolation function, interface gradient energy density, and polycrystalline free energy density, the total free energy equation is obtained:

[0043]

[0044] Where g(η)=η 2 (1-η) 2 κ is the interpolation function; ω is the barrier height; η For the pore phase gradient term coefficient, the last term of the formula represents the interfacial gradient energy density.

[0045] In one implementation method, S3 specifically includes:

[0046] Based on the physical properties of the ceramic nuclear fuel obtained in S3.1, a sintering phase-field model for the ceramic nuclear fuel is established. During the preparation process, a certain number of pores remain in the ceramic nuclear fuel. The size distribution of these residual pores is obtained by simulating the sintering process. The local free energy density function expression in the sintering phase-field model is as follows:

[0047]

[0048] Where C and D are constants; ρ represents the pores during the sintering process, and ρ = 1.0 represents the matrix phase, and ρ = 0.0 represents the porous phase; the first term is the double potential well function, and C determines the barrier height; the second term is the function ξ(φ i The form is as follows:

[0049]

[0050] The total free energy functional F in the sintered phase field model is expressed as:

[0051]

[0052] In the formula, the first term f(ρ, φ) i ) is the local free energy density function, κρ and κ φ These are the coefficients of the energy gradient term;

[0053] S3.2 Assuming the mass is conserved during the sintering process, a translational flux term is introduced into the evolutionary kinetic equation to describe the porosity shrinkage and densification during sintering. Therefore, the evolution equation for the conservative field phase field variable, i.e., the porosity ρ during the sintering process, is:

[0054]

[0055] In the formula: v adv (r) represents the advection velocity field, describing the mass transfer through the motion of local volume elements of a rigid body, in v adv (r) In the calculation, it is assumed that vacancies are the only point defect type present in ceramic nuclear fuel, and grain boundaries and free surfaces act as effective sources and sinks of vacancies;

[0056] φ represents the i-th grain i The evolutionary dynamics equation, after introducing the translational flux term, is expressed as:

[0057]

[0058] In the formula, v advi (r) represents the advection velocity of grain i at position r; L is a coefficient characterizing the grain boundary mobility; the rigid motion of particles during sintering consists of translation and rotation, where the advection velocity field v adv (r) is represented as:

[0059]

[0060] In the formula: v ti (r) represents the translational velocity of the grain with orientation i at position r, expressed as:

[0061]

[0062] V i =∫ V φ i (r)d 3 r (23)

[0063] Where: m t m is the translational migration rate. 2 ·s -1 V i Let be the volume of the particles in grain i; the integral term is the resultant force acting on the center of mass of grain i; κ F ρ0 is the stiffness coefficient of mass density relative to the equilibrium value at the grain boundary; ρ0 is the constant, representing the equilibrium value of the grain boundary mass density; r is the spatial position; and f is the function. v (φ i, φ j This is used to identify the regions between grain boundaries, and its expression is:

[0064]

[0065] In the formula: c* is the grain boundary threshold;

[0066] v ri (r) represents the rotational velocity of grain i at position r, expressed as:

[0067]

[0068] Where: m r m is the rotational mobility. 2 ·s -1 ;T i The torque acting on grain i is expressed as:

[0069]

[0070] In the formula: r ci The position of the centroid of grain i is expressed as:

[0071]

[0072] S3.3 The evolution of pores and grains during the sintering process is obtained by solving the governing equations in the sintering phase field model. The size distribution of the residual pores after sintering is obtained by statistically analyzing ρ. This size distribution is then used as the initial pore distribution for the simulation of the densification phase field of ceramic nuclear fuel irradiation.

[0073] In one implementation method, S4 specifically includes:

[0074] S4.1 The distribution of pore radius during the irradiation densification process of ceramic nuclear fuel is statistically analyzed using the connected region labeling method. The method is as follows: During the connected region labeling process, the initial labeling is performed from top to bottom and from left to right, and the judgment rules are as follows:

[0075] (1). Global marking is performed based on phase field variables. The matrix is ​​marked as invalid and the pores are marked as valid. Each valid pixel is set with an n value.

[0076] (2). When the left and top neighbor pixels of a pixel are invalid, assign a new value n to the pixel, and then n+1;

[0077] (3). When one of the left or top neighbor pixels of a pixel is a valid value, the n value of the valid pixel is assigned to the n value of the pixel.

[0078] (4). When both the left and top neighbor pixels of a pixel are valid values, select the smaller n value and assign it to the n value of that pixel;

[0079] Finally, we will get a matrix consisting of invalid values ​​and a sequence n, where a value m in the sequence n represents the m-th pore; we will then calculate the area of ​​the region contained in each value of n to obtain the area and radius of each pore.

[0080] S4.2 Based on rate theory, the densification mechanism was analyzed to obtain the change law of densification on point defects over time under irradiation conditions;

[0081] The main mechanism of irradiation densification is as follows: if cascade collisions occur close enough to the pores, some of the pore volume will be dispersed into the lattice as vacancies; large pores will be destroyed and their volume will decrease, but most of the vacancies will return to the pores, resulting in minor densification; while small pores will be completely destroyed, forcing vacancies to migrate further, which directly reduces the volume of pores in the fuel.

[0082] If cascade collisions interact with pores, it is assumed that the emitted vacancies remain in the matrix; therefore, the vacancy rate generated by each cascade-pore interaction is:

[0083] K v =2π(r+r) int ) 2 N p Ωf r c si λf sat (28)

[0084] Where r is the radius of the pores, and the statistical method is shown in S4.1; r int N represents the radius of the range of interaction between the cascade process and the pores. p Ω represents the porosity; f represents the lattice volume of the gas atoms; f represents the pore concentration. r c is the fission density; si λ represents the vacancy concentration emitted from pores by a single cascaded stomatal interaction; f represents the fission fragment track length; sat This represents the vacancy saturation.

[0085] In one implementation method, S5 specifically includes:

[0086] S5.1 Establish the relationship between time and fuel consumption. When the fission rate is constant, fuel consumption is directly proportional to time: t = f r ×bu; where t is time, f r Let bu be a constant related to the fission rate, and bu be the burnup. Therefore, the irradiation densification phase field evolution equation, considering the generation and annihilation of point defects under irradiation conditions and the absorption effect of grain boundaries on point defects, is as follows:

[0087] Equation for the evolution of vacancy atom concentration:

[0088]

[0089] Equation of the evolution of interstitial atom concentration:

[0090]

[0091] Equation for the evolution of gas atom concentration:

[0092]

[0093] Evolution equation of porosity phase field variables:

[0094]

[0095] Evolution equation of polycrystalline phase field variables:

[0096]

[0097] In the formula, L is the free interface mobility; M v M i M g These represent the atomic mobilities of interstitial atoms, vacant atoms, and gas atoms, respectively; P v P i P g These represent the production rates of vacancies, interstitial atoms, and gas atoms under irradiation conditions, respectively; R vi For the annihilation terms of point vacancies and interstitial atoms; S v S i This is the absorption term of grain boundaries for vacancies and interstitial atoms; K v Vacancy generation rate under cascade-stomatal interaction

[0098] The expressions for the generation of vacancies and interstitial atoms are as follows:

[0099]

[0100] Where R1 and R2 are two randomly generated numbers between 0 and 1; E is a bias constant representing the varying number of vacancies and interstitial atoms generated by the ex-situ damage; parameter P casc V represents the probability of a cascading collision occurring. G This represents the maximum increase in vacancy concentration caused by a single cascade collision event; the condition η < 0.8 ensures that cascade collisions occur only in the matrix phase and not in the porosiform phase.

[0101] The production rate of gas atoms is:

[0102] P g =2(1-η)2 ΛΩf r Ran (35)

[0103] Where Λ is a constant; Ω is the lattice volume of the gas atoms; f r It is the fission rate; Ran is a random number between 0 and 1;

[0104] When vacancies and interstitial atoms meet, they annihilate and recombine to form a perfect crystal lattice, which can be represented by the following formula:

[0105] R vi =ν r c v c i (36)

[0106] Among them, v r It is the recombination rate: ν r =v b +η 2 v s v b and v s These represent the recombination rates of point defects at the matrix phase and the interface, respectively; v b =4πr iv (D i +D v ) / Ω, r iv It is the composite volume radius; D v and D i These are the diffusion coefficients of vacancies and interstitial atoms, respectively; S v S i The absorption term for vacancies and interstitial atoms at grain boundaries is expressed as follows:

[0107]

[0108] in , which is the grain boundary absorption factor, representing the intensity of absorption of point defects by grain boundaries; It is a function related to the grain boundary position, where Φ < 1.0 at the grain boundary and Φ = 1.0 inside the grain; The equilibrium concentration of vacancies and interstitial atoms;

[0109] S5.2 uses the semi-implicit Fourier spectral method to solve the phase field evolution equation, and further simplifies the formula of the phase field evolution equation to obtain:

[0110]

[0111] Where f is the sum of the volume free energy density and the polycrystalline interaction energy density; taking a Fourier transform of both sides of the equation yields:

[0112]

[0113] In this equation, the variables within parentheses represent the variables after the Fourier transform; k = (k1, k2) are the vector coordinates of the Fourier transform; by implicitly handling linear and second-order operators and explicitly handling other terms, a semi-implicit form of the equation is obtained:

[0114]

[0115] Where Δt is the time step between the nth and (n+1)th steps; further simplifying the equation, we get:

[0116]

[0117] The evolution equations for vacancy concentration, interstitial atom concentration, and gas atom concentration were solved using a similar method, and the results are shown in the following equation:

[0118]

[0119] The equation for the evolution of polycrystalline phase field variables is solved as follows:

[0120]

[0121] In one implementation method, S6 specifically includes:

[0122] S6.1 The numerical solution results in step S5 are visualized using visualization software to obtain the morphological evolution of pores and grains during the densification of ceramic nuclear fuel irradiation:

[0123] The visualization variables characterizing the irradiation densification simulation process are:

[0124]

[0125] In the formula, For defined visualization variables; inside the grain At the grain boundary Inside the pores Therefore, grains, grain boundaries, and pores can be distinguished by differentiating the values ​​of visualized variables at various points in space;

[0126] S6.2 uses the visualization variables obtained in S6.1 to write VTK files and perform visualization processing: The VTK file format contains five basic parts:

[0127] (1). The first part is the file version and identifier;

[0128] (2). The second part is the title;

[0129] (3). The third part is the file format. Enter the format name that describes the file type in this part. The format name is either ASCII or BINARY.

[0130] (4). The fourth part is the structure of the dataset. This part defines the geometric structure of the dataset, including the beginning of the DATASET row and keywords describing the dataset type;

[0131] (5). The fifth part is the attributes of the dataset. This part begins with the keyword POINT_DATA, followed by the number of specified points; then, enter the actual dataset, i.e., the visualization variables obtained in S6.1;

[0132] Finally, import the obtained VTK file into Paraview visualization software for visualization processing;

[0133] S6.3 The porosity is statistically calculated using the simulation results of the phase-field equation. First, the entire simulation region is cyclically processed, and areas where η>0.8 are considered porosity phases and counted. The porosity is then calculated based on the count result N0.

[0134]

[0135] Where, N x and N y These represent the number of grid cells in the x and y directions, respectively.

[0136] In one implementation method, S7 specifically includes:

[0137] S7.1 calculates the effective thermal conductivity during the irradiation densification process of ceramic nuclear fuel. Based on the results of solving the phase field evolution equation in S5, the porosity and grain distribution of the nuclear fuel are obtained. To calculate the effective thermal conductivity under the corresponding microstructure, the temperature distribution related to the microstructure is obtained by solving the steady-state heat conduction equation.

[0138]

[0139] In the formula, κ(η,φ) is the local thermal conductivity, which is related to the heat transfer performance corresponding to the microstructure. It is assumed that from the inside of the grain, κ(η,φ) = κ bulk κ bulk The thermal conductivity is the thermal conductivity under the condition of a complete crystal lattice structure. The presence of grain boundaries will disrupt the integrity of the crystal lattice arrangement and reduce the heat transfer performance. Therefore, it is necessary to define the thermal conductivity at the grain boundary as κ(η,φ)=κ GB κ GB The thermal conductivity at the grain boundary; when located within the pores, κ(η,φ)=κ bubble κ bubbleThe thermal conductivity within the pores is given by the following formula: The steady-state heat conduction equation is solved using the finite difference algorithm to obtain the equilibrium temperature field; the effective thermal conductivity is calculated using the following formula:

[0140]

[0141] Where j Q It is heat flux, T l and T r These are the average temperatures of the left and right boundaries, respectively.

[0142] S7.2 Based on statistical porosity, the irradiation densification process is monitored by tracking the shrinkage of all initial pores over time, as defined below:

[0143]

[0144] Where Sh(t) is the stomatal shrinkage rate at time t; A(0) and A(t) are the stomatal areas at the initial time and time t, respectively.

[0145] Compared with the prior art, the present invention has the following advantages:

[0146] This invention significantly improves the simulation accuracy of the irradiation densification evolution process of ceramic nuclear fuel; it fully considers the influence of various objective conditions on irradiation densification and expands the applicability of phase-field simulation methods.

[0147] Furthermore, the pore size distribution of ceramic nuclear fuel after sintering is obtained through the sintering phase field model and used as an initial condition input into the irradiation densification model to make the simulation process closer to the actual physical process.

[0148] Furthermore, the dynamic equations of point defect evolution during the irradiation densification process are obtained through rate theory, and the irradiation densification process is numerically processed, providing a research method for simulating irradiation densification using the phase field method.

[0149] Furthermore, a semi-implicit Fourier method is used to solve the phase field evolution equation, which improves computational efficiency and accuracy.

[0150] Furthermore, the simulation fully considers the influence of factors such as irradiation-accelerated diffusion, temperature, irradiation intensity, and grain size on the irradiation densification process and its impact on heat transfer performance, thus expanding the applicability of the phase-field simulation method.

[0151] Furthermore, this invention is implemented by programming in the Fortran language and uses software such as Matlab and Paraview for data processing, which has good scalability and flexibility. Attached Figure Description

[0152] Figure 1This is a flowchart of the method of the present invention.

[0153] Figure 2a , Figure 2b , Figure 2c These are the initial microstructure of the sintering process and the simulation time of 40×10. 4 The organizational structure and simulation duration are 100×10 steps. 4 The organizational structure of the steps.

[0154] Figure 3a , Figure 3b , Figure 3c These are the initial microstructure of the irradiation densification process and the simulation duration of 40×10⁻⁶. 4 The organizational structure and simulation duration are 100×10 steps. 4 The organizational structure of the steps.

[0155] Figure 4 for Figure 3a Calculation results of temperature distribution corresponding to the tissue structure.

[0156] Figure 5 This is a graph showing the relationship between density and effective thermal conductivity. Detailed Implementation

[0157] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0158] The main concept of this invention is as follows:

[0159] like Figure 1 As shown, physical property parameters of ceramic nuclear fuel during irradiation were collected, and free energy equations for the evolution of pores and grains during irradiation densification were constructed. The initial pore distribution was obtained by simulating the sintering process of ceramic nuclear fuel and used as the initial condition for irradiation densification simulation. The evolution law of point defects during irradiation densification was obtained through rate theory. Based on the introduction of irradiation conditions and polycrystalline structure, phase-field evolution equations were constructed and solved. The numerical solutions obtained after solving the evolution equations were visualized to obtain the pore morphology and evolution law of the nuclear material during irradiation. The number and size of pores were statistically analyzed to obtain the degree of irradiation densification and densification law. The heat conduction law during nuclear fuel densification was obtained through the statistically obtained pore distribution, thereby enabling the prediction of the service performance of ceramic nuclear fuel.

[0160] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of the invention.

[0161] (1) Select phase field variables and collect nuclear material physical property parameters

[0162] To select phase-field variables that describe the densification process of ceramic nuclear fuel under irradiation, it is necessary to select phase-field variables to represent point defect concentration: vacancy concentration c. v Interstitial atom concentration c i and gas atom concentration c g Simultaneously, a porosity phase sequence parameter η is selected to distinguish between the porosity phase and the matrix phase: η = 1.0 when located in the porosity phase, and η = 0.0 when located in the matrix phase; a set of sequence parameters φ also needs to be selected. i This indicates the grain with index i, and the index parameter indicates whether there is φ in the i-th grain. i =1.0;

[0163] Collect the physical properties of nuclear fuel, including the formation energies of vacancies, interstitial atoms and gas atoms, the diffusion coefficient and gradient coefficient of ceramic nuclear fuel, and the mobility of pores and grain interfaces.

[0164] Based on the diffusion coefficients of vacancies, interstitial atoms, and gas atoms, the mobility considering point defect irradiation-accelerated diffusion is calculated. The diffusion coefficient of atoms in the crystal matrix is ​​expressed as:

[0165]

[0166] Where θ is a material-related constant; ν is the vibrational frequency of the atom; G m k is the free energy required for an atom to migrate from its equilibrium position to its nearest position. B Here, c is Boltzmann's constant, T is absolute temperature; v Let be the vacancy concentration; then the vacancy diffusion coefficient after irradiation acceleration is written as:

[0167]

[0168] Where c v e c is the thermal equilibrium concentration of vacancies. v r The vacancy concentration introduced by irradiation; D0 is the pre-factor of the diffusion coefficient; E i To activate energy;

[0169] Vacancy mobility M v The following expression exists:

[0170]

[0171] Where D v To account for the vacancy diffusion coefficient after irradiation acceleration; V m R is the lattice volume; R is the ideal gas constant.

[0172] (2) Constructing the free energy equation

[0173] Based on the derivation of materials thermodynamics theory, the free energy density function of the ceramic nuclear fuel matrix phase is calculated using the following formula:

[0174]

[0175] Among them, c v m c i m c g m These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the matrix phase, respectively. These represent the formation energies of vacancies, interstitial atoms, and gas atoms, respectively; k B Boltzmann constant; T is absolute temperature;

[0176] The free energy density function of the porous phase of ceramic nuclear fuel is as follows:

[0177]

[0178] Among them, c v b c i b c g b These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the porous phase, respectively, where A and B are constants, and f is the concentration of gas atoms in the porous phase. g b (c g b Let f be the gas Gibbs free energy function related to the gas atom concentration, obtained through reference thermophysics. g b (c g b The expression for ) is:

[0179]

[0180] Among them, V a The volume of a single U atom; n Q b is the quantum concentration; b is the ideal gas constant.

[0181] Based on the obtained free energy density functions of the matrix phase and the porosity phase, an interpolation function is introduced to obtain the bulk free energy density function:

[0182]

[0183] Where h(η) = η 3 (6η 2The interpolation function (-15η+10) is constructed. When η = 1.0, it satisfies h(η) = 1.0; and when η = 0.0, it satisfies h(η) = 0.0. The relationship between the total defect concentration and the defect concentration in each phase is as follows:

[0184]

[0185]

[0186] When ceramic nuclear fuel reaches equilibrium, the chemical potentials corresponding to the matrix phase and the pores are equal, therefore:

[0187]

[0188] The polycrystalline free energy density function is:

[0189]

[0190] Where α, β, γ, and δ are phenomenological parameters; φ i Indicates the grain with serial number i; φ j This indicates the grain with serial number j;

[0191] Combining bulk free energy density, interpolation function, interface gradient energy density, and polycrystalline free energy density, the total free energy equation is obtained:

[0192]

[0193] Where g(η)=η 2 (1-η) 2 κ is the interpolation function; ω is the barrier height; η For the pore phase gradient term coefficient, the last term of the formula represents the interfacial gradient energy density.

[0194] (3) Obtain the initial pore size distribution by combining the sintering phase field model.

[0195] Based on the physical property parameters of the ceramic nuclear fuel obtained in step (1), a sintering phase-field model of the ceramic nuclear fuel is established: a certain number of pores remain during the preparation process of the ceramic nuclear fuel, and the size distribution of the residual pores is obtained by simulating the sintering process; the expression of the local free energy density function in the sintering phase-field model is as follows:

[0196]

[0197] Where C and D are constants; ρ represents the pores during the sintering process, and ρ = 1.0 represents the matrix phase, and ρ = 0.0 represents the porous phase; the first term is the double potential well function, and C determines the barrier height; the second term is the function ξ(φ i The form is as follows:

[0198]

[0199] The total free energy functional F in the sintered phase field model is expressed as:

[0200]

[0201] In the formula, the first term f(ρ, φ) i ) is the local free energy density function, κ ρ and κ φ These are the coefficients of the energy gradient term;

[0202] Assuming mass is conserved during sintering, a translational flux term is introduced into the evolutionary kinetic equation to describe the porosity shrinkage and densification during sintering. Therefore, the evolution equation for the conservative field phase field variable, i.e., the porosity ρ during sintering, is as follows:

[0203]

[0204] In the formula: v adv (r) represents the advection velocity field, describing the mass transfer through the motion of local volume elements of a rigid body, in v adv (r) In the calculation, it is assumed that vacancies are the only point defect type present in ceramic nuclear fuel, and grain boundaries and free surfaces act as effective sources and sinks of vacancies;

[0205] φ represents the i-th grain i The evolutionary dynamics equation, after introducing the translational flux term, is expressed as:

[0206]

[0207] In the formula, v advi (r) represents the advection velocity of grain i at position r; L is a coefficient characterizing the grain boundary mobility; the rigid motion of particles during sintering consists of translation and rotation, where the advection velocity field v adv (r) is represented as:

[0208]

[0209] In the formula: v ti (r) represents the translational velocity of the grain with orientation i at position r, expressed as:

[0210]

[0211] V i =∫ V φ i (r)d 3 r (23)

[0212] Where: m t m is the translational migration rate.2 ·s -1 V i Let be the volume of the particles in grain i; the integral term is the resultant force acting on the center of mass of grain i; κ F ρ0 is the stiffness coefficient of mass density relative to the equilibrium value at the grain boundary; ρ0 is the constant, representing the equilibrium value of the grain boundary mass density; r is the spatial position; and f is the function. v (φ i , φ j This is used to identify the regions between grain boundaries, and its expression is:

[0213]

[0214] In the formula: c* is the grain boundary threshold;

[0215] v ri (r) represents the rotational velocity of grain i at position r, expressed as:

[0216]

[0217] Where: m r m is the rotational mobility. 2 ·s -1 ;T i The torque acting on grain i is expressed as:

[0218]

[0219] In the formula: r ci The position of the centroid of grain i is expressed as:

[0220]

[0221] The evolution of pores and grains during the sintering process is obtained by solving the governing equations in the sintering phase field model. The size distribution of residual pores after sintering is obtained by statistically analyzing ρ. This size distribution is then used as the initial pore distribution for the simulation of the densification phase field of ceramic nuclear fuel irradiation.

[0222] (4) Obtain the law of defect evolution through rate theory

[0223] The distribution of pore radius during the irradiation densification process of ceramic nuclear fuel was statistically analyzed using the connected region labeling method. The method is as follows: During the connected region labeling process, the initial labeling is performed from top to bottom and from left to right, and the judgment rules are as follows:

[0224] (1). Global marking is performed based on phase field variables. The matrix is ​​marked as invalid and the pores are marked as valid. Each valid pixel is set with an n value.

[0225] (2). When the left and top neighbor pixels of a pixel are invalid, assign a new value n to the pixel, and then n+1;

[0226] (3). When one of the left or top neighbor pixels of a pixel is a valid value, the n value of the valid pixel is assigned to the n value of the pixel.

[0227] (4). When both the left and top neighbor pixels of a pixel are valid values, select the smaller n value and assign it to the n value of that pixel;

[0228] Finally, we will get a matrix consisting of invalid values ​​and a sequence n, where a value m in the sequence n represents the m-th pore; we will then calculate the area of ​​the region contained in each value of n to obtain the area and radius of each pore.

[0229] Based on the rate theory analysis of the densification mechanism, the variation law of densification of point defects with time under irradiation conditions was obtained.

[0230] The main mechanism of irradiation densification is as follows: if cascade collisions occur close enough to the pores, some of the pore volume will be dispersed into the lattice as vacancies; large pores will be destroyed and their volume will decrease, but most of the vacancies will return to the pores, resulting in minor densification; while small pores will be completely destroyed, forcing vacancies to migrate further, which directly reduces the volume of pores in the fuel.

[0231] If cascade collisions interact with pores, it is assumed that the emitted vacancies remain in the matrix; therefore, the vacancy rate generated by each cascade-pore interaction is:

[0232] K v =2π(r+r) int ) 2 N p Ωf r c si λf sat (28)

[0233] Where r is the radius of the pores, and the statistical method is shown in S4.1; r int N represents the radius of the range of interaction between the cascade process and the pores. p Ω represents the porosity; f represents the lattice volume of the gas atoms; f represents the pore concentration. r c is the fission density; si λ represents the vacancy concentration emitted from pores by a single cascaded stomatal interaction; f represents the fission fragment track length; sat This represents the vacancy saturation.

[0234] (5) Establish and solve the phase field evolution equation.

[0235] Establish the relationship between time and fuel consumption. When the fission rate is constant, fuel consumption is directly proportional to time: t = f r ×bu; where t is time, f r Let bu be a constant related to the fission rate, and bu be the burnup. Therefore, the irradiation densification phase field evolution equation, considering the generation and annihilation of point defects under irradiation conditions and the absorption effect of grain boundaries on point defects, is as follows:

[0236] Equation for the evolution of vacancy atom concentration:

[0237]

[0238] Equation of the evolution of interstitial atom concentration:

[0239]

[0240] Equation for the evolution of gas atom concentration:

[0241]

[0242] Evolution equation of porosity phase field variables:

[0243]

[0244] Evolution equation of polycrystalline phase field variables:

[0245]

[0246] In the formula, L is the free interface mobility; M v M i M g These represent the atomic mobilities of interstitial atoms, vacant atoms, and gas atoms, respectively; P v P i P g These represent the production rates of vacancies, interstitial atoms, and gas atoms under irradiation conditions, respectively; R vi For the annihilation terms of point vacancies and interstitial atoms; S v S i This is the absorption term of grain boundaries for vacancies and interstitial atoms; K v The vacancy generation rate under cascade-stomatal interaction.

[0247] The expressions for the generation of vacancies and interstitial atoms are as follows:

[0248]

[0249] Where R1 and R2 are two randomly generated numbers between 0 and 1; E is a bias constant representing the varying number of vacancies and interstitial atoms generated by the ex-situ damage; parameter P cascV represents the probability of a cascading collision occurring. G This represents the maximum increase in vacancy concentration caused by a single cascade collision event; the condition η < 0.8 ensures that cascade collisions occur only in the matrix phase and not in the porosiform phase.

[0250] The production rate of gas atoms is:

[0251] P g =2(1-η) 2 ΛΩf r Ran (35)

[0252] Where Λ is a constant; Ω is the lattice volume of the gas atoms; f r It is the fission rate; Ran is a random number between 0 and 1;

[0253] When vacancies and interstitial atoms meet, they annihilate and recombine to form a perfect crystal lattice, which can be represented by the following formula:

[0254] R vi =ν r c v c i (36)

[0255] Among them, v r It is the recombination rate: ν r =v b +η 2 v s v b and v s These represent the recombination rates of point defects at the matrix phase and the interface, respectively; v b =4πr iv (D i +D v ) / Ω, r iv It is the composite volume radius; D v and D i These are the diffusion coefficients of vacancies and interstitial atoms, respectively; S v S i The absorption term for vacancies and interstitial atoms at grain boundaries is expressed as follows:

[0256]

[0257] in , which is the grain boundary absorption factor, representing the intensity of absorption of point defects by grain boundaries; It is a function related to the grain boundary position, where Φ < 1.0 at the grain boundary and Φ = 1.0 inside the grain; The equilibrium concentration of vacancies and interstitial atoms;

[0258] The phase field evolution equation is solved using the semi-implicit Fourier spectral method, and the formula for the phase field evolution equation is further simplified to obtain:

[0259]

[0260] Where f is the sum of the volume free energy density and the polycrystalline interaction energy density; taking a Fourier transform of both sides of the equation yields:

[0261]

[0262] In this equation, the variables within parentheses represent the variables after the Fourier transform; k = (k1, k2) are the vector coordinates of the Fourier transform; by implicitly handling linear and second-order operators and explicitly handling other terms, a semi-implicit form of the equation is obtained:

[0263]

[0264] Where Δt is the time step between the nth and (n+1)th steps; further simplifying the equation, we get:

[0265]

[0266] The evolution equations for vacancy concentration, interstitial atom concentration, and gas atom concentration were solved using a similar method, and the results are shown in the following equation:

[0267]

[0268] The equation for the evolution of polycrystalline phase field variables is solved as follows:

[0269]

[0270] (6) Visualization and simulation results statistics

[0271] The numerical solution results in step S5 are visualized using visualization software to obtain the morphological evolution of pores and grains during the densification of ceramic nuclear fuel irradiation:

[0272] The visualization variables characterizing the irradiation densification simulation process are:

[0273]

[0274] In the formula, For defined visualization variables; inside the grain At the grain boundary Inside the pores Therefore, grains, grain boundaries, and pores can be distinguished by differentiating the values ​​of visualized variables at various points in space;

[0275] Use the visualization variables obtained in step (6) to write a VTK file and perform visualization processing: The VTK file format contains five basic parts:

[0276] (1). The first part is the file version and identifier;

[0277] (2). The second part is the title;

[0278] (3). The third part is the file format. Enter the format name that describes the file type in this part. The format name is either ASCII or BINARY.

[0279] (4). The fourth part is the structure of the dataset. This part defines the geometric structure of the dataset, including the beginning of the DATASET row and keywords describing the dataset type;

[0280] (5). The fifth part is the attributes of the dataset. This part begins with the keyword POINT_DATA, followed by the number of specified points; then, enter the actual dataset, i.e., the visualization variables obtained in S6.1;

[0281] Finally, import the obtained VTK file into Paraview visualization software for visualization processing;

[0282] Porosity is statistically analyzed using simulation results from the phase-field equation. First, the entire simulation region is cyclically processed, and areas where η > 0.8 are considered porosity phases and counted. The porosity is then calculated based on the count result N0.

[0283]

[0284] Where, N x and N y These represent the number of grid cells in the x and y directions, respectively.

[0285] (7) Calculate the effective thermal conductivity and the densification shrinkage rate.

[0286] The effective thermal conductivity during the irradiation densification process of ceramic nuclear fuel was calculated. Based on the results of solving the phase field evolution equation using S5, the porosity and grain distribution of the nuclear fuel were obtained. To calculate the effective thermal conductivity under the corresponding microstructure, the temperature distribution related to the microstructure was obtained by solving the steady-state heat conduction equation.

[0287]

[0288] In the formula, κ(η,φ) is the local thermal conductivity, which is related to the heat transfer performance corresponding to the microstructure. It is assumed that from the inside of the grain, κ(η,φ) = κ bulk κ bulkThe thermal conductivity is the thermal conductivity under the condition of a complete crystal lattice structure. The presence of grain boundaries will disrupt the integrity of the crystal lattice arrangement and reduce the heat transfer performance. Therefore, it is necessary to define the thermal conductivity at the grain boundary as κ(η,φ)=κ GB κ GB The thermal conductivity at the grain boundary; when located within the pores, κ(η,φ)=κ bubble κ bubble The thermal conductivity within the pores is given by the following formula: The steady-state heat conduction equation is solved using the finite difference algorithm to obtain the equilibrium temperature field; the effective thermal conductivity is calculated using the following formula:

[0289]

[0290] Where j Q It is heat flux, T l and T r These are the average temperatures of the left and right boundaries, respectively.

[0291] The irradiation densification process is monitored by tracking the shrinkage of all initial pores over time based on statistical porosity, as defined below:

[0292]

[0293] Where Sh(t) is the stomatal shrinkage rate at time t; A(0) and A(t) are the stomatal areas at the initial time and time t, respectively.

[0294] The following are some specific examples.

[0295] Example 1

[0296] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0297] This embodiment provides a method for determining the effect of the degree of irradiation densification of uranium dioxide nuclear fuel on its thermal conductivity. According to the process in the specific implementation, the sintering process of uranium dioxide at 1273K was numerically simulated using a phase-field model.

[0298] First, a phase-field model for uranium dioxide nuclear fuel sintering is established. The free energy expression in the sintering phase-field model is as follows:

[0299]

[0300] Where C and D are constants; ρ represents the pores during the sintering process, with ρ = 1.0 representing the matrix phase and ρ = 0.0 representing the porous phase; the first term is the double-well function, where C determines the barrier height. The second term is the function ξ(φ). i The form is as follows:

[0301]

[0302] The total free energy functional in the sintering phase-field model is expressed as:

[0303]

[0304] In the formula, the first term f is the local free energy density function, κ ρ and κ φ These are the coefficients of the energy gradient term;

[0305] Assuming mass is conserved during sintering, a translational flux term is introduced into the evolutionary kinetic equations to describe the porosity shrinkage and densification during sintering. Therefore, the evolution equation for the conservative field phase variable ρ is:

[0306]

[0307] In the formula: v adv (r) represents the advection velocity field, describing mass transfer through the motion of local volume elements of a rigid body. In v adv (r) In the calculation, it is assumed that vacancies are the only point defect type that exists in the system, and grain boundaries and free surfaces act as effective sources and sinks of vacancies;

[0308] Non-conservative phase field variable φ i The evolutionary dynamics equation, after introducing the translational flux term, is expressed as:

[0309]

[0310] In the formula, v advi (r) represents the advection velocity of grain i at position r; L is a coefficient characterizing the grain boundary mobility; the rigid motion of particles during sintering consists of translation and rotation, where the advection velocity field v adv (r) is represented as:

[0311]

[0312] In the formula: v ti (r) represents the translational velocity of the grain with orientation i at position r, expressed as:

[0313]

[0314] V i =∫ V φ i (r)d 3 r (59)

[0315] Where: m t m is the translational migration rate. 2 ·s -1 Vi Let κ be the volume of the particle with orientation i; the integral term is the resultant force acting on the centroid of grain i; F ρ0 is the stiffness coefficient of mass density relative to the equilibrium value at the grain boundary; ρ0 is the constant, representing the equilibrium value of the grain boundary mass density; r is the spatial position; and f is the function. v The expression used to identify regions between grain boundaries is:

[0316]

[0317] In the formula: c* is the grain boundary threshold; v ri (r) represents the rotational velocity of the grain with orientation i at position r, expressed as:

[0318]

[0319] Where: m r m is the rotational mobility. 2 ·s -1 ;T i The torque acting on particle i is expressed as:

[0320]

[0321] In the formula: r ci The position of the centroid of grain i is expressed as:

[0322]

[0323] The phase-field evolution equation was solved using a semi-implicit Fourier method, and the numerical results were visualized using Paraview. The microstructure of the sintering process in the simulation results is shown below. Figure 2a , Figure 2b , Figure 2c As shown: Figure 2a For the initial organizational structure, Figure 2b To simulate a duration of 40×10 4 The corresponding organizational structure, Figure 2c To simulate a duration of 100×10 4 The corresponding microstructure was determined through simulation calculations in Example 1. The pore distribution during the sintering process was obtained, providing an initial microstructure for irradiation densification.

[0324] Example 2

[0325] This example, based on Example 1, conducts a phase-field simulation study of the uranium dioxide irradiation densification process. First, a free energy equation is constructed. Based on thermodynamic theory, the free energy density function of the nuclear fuel matrix phase can be calculated using the following formula:

[0326]

[0327] Among them, c v m c i m c g m These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the matrix phase, respectively. These represent the formation energies of vacancies, interstitial atoms, and gas atoms, respectively; k B Boltzmann constant; T is absolute temperature;

[0328] The free energy density function of the porous phase of S2.2 nuclear fuel is as follows:

[0329]

[0330] Among them, c v b c i b c g b These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the porous phase, respectively, where A and B are constants, and f is the concentration of gas atoms in the porous phase. g b (c g Let f(cgb) be the gas Gibbs free energy function related to the concentration of gas atoms. Using reference thermophysics, the expression for the function fgb(cgb) can be obtained:

[0331]

[0332] Among them, V a The volume of a single U atom; n Q b is the quantum concentration; b is the ideal gas constant.

[0333] S2.3 Based on the obtained free energy density functions of the matrix phase and the porosity phase, an interpolation function is introduced to obtain the bulk free energy density function:

[0334]

[0335] Where h(η) = η 3 (6η 2 The interpolation function (-15η+10) is constructed. When η = 1.0, it satisfies h(η) = 1.0; and when η = 0.0, it satisfies h(η) = 0.0. The relationship between the total defect concentration and the defect concentration in each phase is as follows:

[0336]

[0337] When the system reaches equilibrium, the chemical potentials corresponding to the matrix phase and the pores are equal, therefore:

[0338]

[0339] The polycrystalline free energy density function is:

[0340]

[0341] Where α, β, γ and δ are phenomenological parameters;

[0342] Combining bulk free energy density, interpolation function, interface gradient energy density, and polycrystalline free energy density, the total free energy equation is obtained:

[0343]

[0344] Where g(η)=η 2 (1-η) 2 κ is the interpolation function; ω is the barrier height; η For the pore phase gradient term coefficient, the last term of the formula represents the gradient energy density.

[0345] If cascade collisions interact with pores, it can be assumed that the emitted vacancies remain in the matrix. Therefore, the vacancy rate generated by each cascade-pore interaction is:

[0346] K v =2π(r+r) int ) 2 N p Ωf r c si λf sat (76)

[0347] Where r is the radius of the pores, and the statistical method is shown in S4.1; r int N represents the radius of the range of interaction between the cascade process and the pores. p f is the stomatal concentration. r c is the fission density; si λ represents the vacancy concentration emitted from pores by a single cascaded stomatal interaction; f represents the fission fragment track length; sat This represents the vacancy saturation.

[0348] Establish the relationship between time and fuel consumption. When the fission rate is constant, fuel consumption is directly proportional to time: t = f r ×bu; where t is time, f r Let bu be a constant related to the fission rate, and bu be the burnup. Therefore, the irradiation densification phase field evolution equation, considering the generation and annihilation of point defects under irradiation conditions and the absorption effect of grain boundaries on point defects, is as follows:

[0349] Equation for the evolution of vacancy atom concentration:

[0350]

[0351] Equation of the evolution of interstitial atom concentration:

[0352]

[0353] Equation for the evolution of gas atom concentration:

[0354]

[0355] Evolution equation of porosity phase field variables:

[0356]

[0357] Evolution equation of polycrystalline phase field variables:

[0358]

[0359] In the formula, t is the simulation time; L is the free interface mobility; M v M i M g These represent the atomic mobilities of interstitial atoms, vacant atoms, and gas atoms, respectively; P v P i P g These represent the production rates of vacancy and interstitial gas atoms under irradiation conditions; R vi For the annihilation terms of point vacancies and interstitial atoms; S v S i This is the absorption term of grain boundaries for vacancies and interstitial atoms;

[0360] P v P i and P g Let represent the production rates of gas atoms, vacancies, and interstitial atoms under irradiation conditions, respectively; the expressions for the production rates of vacancies and interstitial atoms are as follows:

[0361]

[0362] Where R1 and R2 are two randomly generated numbers between 0 and 1; E is a bias constant representing the varying number of vacancies and interstitial atoms generated by the ex-situ damage; parameter P casc V represents the probability of a cascading collision occurring. G This represents the maximum increase in vacancy concentration caused by a single cascade collision event; the condition η < 0.8 ensures that cascade collisions occur only in the matrix phase and not in the porosiform phase.

[0363] The production rate of gas atoms is:

[0364] P g =2(1-η)2 ΛΩf r Ran (91)

[0365] Where Λ is a constant; Ω is the lattice volume of the gas atoms; f r It is the fission rate; Ran is a random number between 0 and 1;

[0366] When vacancies and interstitial atoms meet, they annihilate and recombine to form a perfect crystal lattice, which can be represented by the following formula:

[0367] R vi =ν r c v c i (92)

[0368] Among them, v r It is the recombination rate: ν r =v b +η 2 v s v b and v s These represent the recombination rates of point defects at the matrix phase and the interface, respectively; v b =4πr iv (D i +D v ) / Ω, riv is the composite volume radius; Dv and Di are the diffusion coefficients of vacancies and interstitial atoms, respectively; Ω is the lattice volume; S v S i The absorption term for vacancies and interstitial atoms at grain boundaries is expressed as follows:

[0369]

[0370] in , which is the grain boundary absorption factor, representing the intensity of absorption of point defects by grain boundaries; It is a function related to the grain boundary position, where Φ < 1.0 at the grain boundary and Φ = 1.0 inside the grain; The equilibrium concentration of vacancies and interstitial atoms;

[0371] The phase field evolution equation is solved using the semi-implicit Fourier method, and the formula for the phase field evolution equation is further simplified to obtain:

[0372]

[0373] Where f is the sum of the volume free energy density and the polycrystalline interaction energy density; taking a Fourier transform of both sides of the equation yields:

[0374]

[0375] Here, the variables within parentheses represent the variables after the Fourier transform; k = (k1, k2) are the vector coordinates of the Fourier transform; by implicitly handling linear and second-order operators and explicitly handling other terms, a semi-implicit form of the equation can be obtained:

[0376]

[0377] Where Δt is the time step between the nth and (n+1)th steps; further simplifying the equation, we get:

[0378]

[0379] The evolution equations for vacancy concentration, interstitial atom concentration, and gas atom concentration can be solved using a similar method, and the results are shown in the following equation:

[0380]

[0381] The equation for the evolution of polycrystalline phase field variables is solved as follows:

[0382]

[0383] The phase-field evolution equation was solved using a semi-implicit Fourier method, and the numerical results were visualized using Paraview. The irradiation densification process for different simulation durations is shown below. Figure 3a , Figure 3b , Figure 3c As shown, pores shrink over time. Furthermore, the change in thermal conductivity corresponding to the irradiation densification process was calculated, where... Figure 4 for Figure 3a The corresponding temperature distribution calculation results, Figure 5 The graph shows the relationship between thermal conductivity and effective thermal conductivity. Thermal conductivity increases with increasing irradiation density.

Claims

1. A method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on its thermal conductivity, characterized in that: Includes the following steps: S1 selects the phase field variables that describe the densification process of ceramic nuclear fuel under irradiation; Collect physical properties of ceramic nuclear fuel, including the formation energies of vacancies, interstitial atoms and gas atoms, the diffusion coefficient and gradient coefficient of ceramic nuclear fuel after irradiation acceleration, and the mobility of pores and grain interfaces. S2 obtains the free energy density function of the ceramic nuclear fuel matrix phase based on the material thermodynamics theory, according to the physical property parameters of the ceramic nuclear fuel collected in step S1. By combining the phase field empirical formula, the free energy density function of the porous phase in the ceramic nuclear fuel is obtained. Then, the total free energy equation is obtained by combining the polycrystalline interaction energy and the interface gradient energy. S3 establishes a sintering phase field model that considers the rigid motion of ceramic nuclear fuel particles, simulates the physical process of sintering ceramic nuclear fuel particles into nuclear fuel, obtains the pore distribution at the moment when the sintering preparation of ceramic nuclear fuel is completed, and uses this pore distribution as the initial structure for the simulation of the irradiation densification phase field of ceramic nuclear fuel. S4 uses the connected region labeling method to statistically analyze the pore radius distribution during the irradiation densification process of ceramic nuclear fuel; then, combined with rate theory, it analyzes the densification mechanism and obtains the time-dependent variation law of point defects in densification under irradiation conditions. Based on the total free energy equation obtained in step S2 and the point defect dynamics equation obtained in step S4, and taking into account the generation and annihilation of point defects under irradiation conditions and the absorption of point defects by grain boundaries, S5 constructs an irradiation densification phase field evolution equation that varies with burnup; the semi-implicit Fourier spectroscopy method is used to solve the phase field evolution equation to obtain a numerical solution. S6 visualizes the numerical solution obtained in step S5 to obtain the morphology of the initial pore compaction process during the compaction of ceramic nuclear fuel; and statistically analyzes the porosity at different times to obtain the compaction law of ceramic nuclear fuel irradiation. Based on the obtained pore distribution and porosity, S7 coupled the phase field-based thermal conductivity calculation formula to obtain the thermal conductivity variation law during the irradiation densification process of ceramic nuclear fuel.

2. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S1 includes the following sub-steps: S1.1 Selecting phase field variables to describe the densification process of ceramic nuclear fuel under irradiation requires selecting phase field variables to represent point defect concentration: vacancy concentration c. v Interstitial atom concentration c i and gas atom concentration c g Simultaneously, a porosity phase sequence parameter η is selected to distinguish between the porosity phase and the matrix phase: η = 1.0 when located in the porosity phase, and η = 0.0 when located in the matrix phase; a set of sequence parameters φ also needs to be selected. i This indicates the grain with index i, and the index parameter indicates whether there is φ in the i-th grain. i =1.0; S1.2 Collect the physical properties of nuclear fuel, including the formation energies of vacancies, interstitial atoms and gas atoms, the diffusion coefficient and gradient coefficient of ceramic nuclear fuel, and the mobility of pores and grain interfaces. S1.3 Based on the diffusion coefficients of vacancies, interstitial atoms, and gas atoms in S1.2, calculate the mobility considering point defect irradiation-accelerated diffusion. The diffusion coefficient of atoms in the crystal matrix is ​​expressed as: Where θ is a material-related constant; ν is the vibrational frequency of the atom; G m k is the free energy required for an atom to migrate from its equilibrium position to its nearest position. B Here, c is Boltzmann's constant, T is absolute temperature; v Let be the vacancy concentration; then the vacancy diffusion coefficient after irradiation acceleration is written as: Where c v e c is the thermal equilibrium concentration of vacancies. v r The vacancy concentration introduced by irradiation; D0 is the pre-factor of the diffusion coefficient; E i To activate energy; Vacancy mobility M v The following expression exists: Where D v To account for the vacancy diffusion coefficient after irradiation acceleration; V m R is the lattice volume; R is the ideal gas constant.

3. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S2 includes the following sub-steps: Based on the thermodynamics of materials, the free energy density function of the ceramic nuclear fuel matrix phase is calculated using the following formula: Among them, c v m c i m c g m These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the matrix phase, respectively. These represent the formation energies of vacancies, interstitial atoms, and gas atoms, respectively; k B Boltzmann constant; T is absolute temperature; The free energy density function of the porous phase of S2.2 ceramic nuclear fuel is as follows: Among them, c v b c i b c g b These represent the concentrations of vacancies, interstitial atoms, and gas atoms in the porous phase, respectively, where A and B are constants, and f is the concentration of gas atoms in the porous phase. g b (c g b Let f be the gas Gibbs free energy function related to the gas atom concentration, obtained through reference thermophysics. g b (c g b The expression for ) is: Among them, V a The volume of a single U atom; n Q b is the quantum concentration; b is the ideal gas constant. S2.3 Based on the obtained free energy density functions of the matrix phase and the porosity phase, an interpolation function is introduced to obtain the bulk free energy density function: Where h(η) = η 3 (6η 2 The interpolation function (-15η+10) is constructed. When η = 1.0, it satisfies h(η) = 1.0; and when η = 0.0, it satisfies h(η) = 0.

0. The relationship between the total defect concentration and the defect concentration in each phase is as follows: When ceramic nuclear fuel reaches equilibrium, the chemical potentials corresponding to the matrix phase and the pores are equal, therefore: The polycrystalline free energy density function is: Where α, β, γ, and δ are phenomenological parameters; φ i Indicates the grain with serial number i; φ j This indicates the grain with serial number j; Combining bulk free energy density, interpolation function, interface gradient energy density, and polycrystalline free energy density, the total free energy equation is obtained: Where g(η)=η 2 (1-η) 2 κ is the interpolation function; ω is the barrier height; η For the pore phase gradient term coefficient, the last term of the formula represents the interfacial gradient energy density.

4. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S3 includes the following sub-steps: Based on the physical properties of the ceramic nuclear fuel obtained in S3.1, a sintering phase-field model for the ceramic nuclear fuel is established. During the preparation process, a certain number of pores remain in the ceramic nuclear fuel. The size distribution of these residual pores is obtained by simulating the sintering process. The local free energy density function expression in the sintering phase-field model is as follows: Where C and D are constants; ρ represents the pores during the sintering process, and ρ = 1.0 represents the matrix phase, and ρ = 0.0 represents the porous phase; the first term is the double potential well function, and C determines the barrier height; the second term is the function ξ(φ i The form is as follows: The total free energy functional F in the sintered phase field model is expressed as: In the formula, the first term f(ρ, φ) i ) is the local free energy density function, κ ρ and κ φ These are the coefficients of the energy gradient term; S3.2 Assuming the mass is conserved during the sintering process, a translational flux term is introduced into the evolutionary kinetic equation to describe the porosity shrinkage and densification during sintering. Therefore, the evolution equation for the conservative field phase field variable, i.e., the porosity ρ during the sintering process, is: In the formula: v adv (r) represents the advection velocity field, describing the mass transfer through the motion of local volume elements of a rigid body, in v adv (r) In the calculation, it is assumed that vacancies are the only point defect type present in ceramic nuclear fuel, and grain boundaries and free surfaces act as effective sources and sinks of vacancies; φ represents the i-th grain i The evolutionary dynamics equation, after introducing the translational flux term, is expressed as: In the formula, v advi (r) represents the advection velocity of grain i at position r; L is a coefficient characterizing the grain boundary mobility; the rigid motion of particles during sintering consists of translation and rotation, where the advection velocity field v adv (r) is represented as: In the formula: v ti (r) represents the translational velocity of the grain with orientation i at position r, expressed as: Where: m t m is the translational migration rate. 2 ·s -1 V i Let be the volume of the particles in grain i; the integral term is the resultant force acting on the center of mass of grain i; κ F ρ0 is the stiffness coefficient of mass density relative to the equilibrium value at the grain boundary; ρ0 is the constant, representing the equilibrium value of the grain boundary mass density; r is the spatial position; and f is the function. v (φ i , φ j This is used to identify the regions between grain boundaries, and its expression is: In the formula: c* is the grain boundary threshold; v ri (r) represents the rotational velocity of grain i at position r, expressed as: Where: m r m is the rotational mobility. 2 ·s -1 ;T i The torque acting on grain i is expressed as: In the formula: r ci The position of the centroid of grain i is expressed as: S3.3 The evolution of pores and grains during the sintering process is obtained by solving the governing equations in the sintering phase field model. The size distribution of the residual pores after sintering is obtained by statistically analyzing ρ. This size distribution is then used as the initial pore distribution for the simulation of the densification phase field of ceramic nuclear fuel irradiation.

5. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S4 includes the following sub-steps: S4.1 The distribution of pore radius during the irradiation densification process of ceramic nuclear fuel is statistically analyzed using the connected region labeling method. The method is as follows: During the connected region labeling process, the initial labeling is performed from top to bottom and from left to right, and the judgment rules are as follows: (1). Global marking is performed based on phase field variables. The matrix is ​​marked as invalid and the pores are marked as valid. Each valid pixel is set with an n value. (2). When the left and top neighbor pixels of a pixel are invalid, assign a new value n to the pixel, and then n+1; (3). When one of the left or top neighbor pixels of a pixel is a valid value, the n value of the valid pixel is assigned to the n value of the pixel. (4). When both the left and top neighbor pixels of a pixel are valid values, select the smaller n value and assign it to the n value of that pixel; Finally, we will get a matrix consisting of invalid values ​​and a sequence n, where a value m in the sequence n represents the m-th pore; we will then calculate the area of ​​the region contained in each value of n to obtain the area and radius of each pore. S4.2 Based on rate theory, the densification mechanism was analyzed to obtain the change law of densification on point defects over time under irradiation conditions; The main mechanism of irradiation densification is as follows: if cascade collisions occur close enough to the pores, some of the pore volume will be dispersed into the lattice as vacancies; large pores will be destroyed and their volume will decrease, but most of the vacancies will return to the pores, resulting in minor densification; while small pores will be completely destroyed, forcing vacancies to migrate further, which directly reduces the volume of pores in the fuel. If cascade collisions interact with pores, it is assumed that the emitted vacancies remain in the matrix; therefore, the vacancy rate generated by each cascade-pore interaction is: K v =2π(r+r int ) 2 N p Ωf r c si λf sat (28) Where r is the radius of the pores, and the statistical method is shown in S4.1; r int N represents the radius of the range of interaction between the cascade process and the pores. p Ω represents the porosity; f represents the lattice volume of the gas atoms; f represents the pore concentration. r c is the fission density; si λ represents the vacancy concentration emitted from pores by a single cascaded stomatal interaction; f represents the fission fragment track length; sat This represents the vacancy saturation.

6. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S5 includes the following sub-steps: S5.1 Establish the relationship between time and fuel consumption. When the fission rate is constant, fuel consumption is directly proportional to time: t = f r ×bu; where t is time, f r Let bu be a constant related to the fission rate, and bu be the burnup. Therefore, the irradiation densification phase field evolution equation, considering the generation and annihilation of point defects under irradiation conditions and the absorption effect of grain boundaries on point defects, is as follows: Equation for the evolution of vacancy atom concentration: Equation of the evolution of interstitial atom concentration: Equation for the evolution of gas atom concentration: Evolution equation of porosity phase field variables: Evolution equation of polycrystalline phase field variables: In the formula, L is the free interface mobility; M v M i M g These represent the atomic mobilities of interstitial atoms, vacant atoms, and gas atoms, respectively; P v P i P g These represent the production rates of vacancies, interstitial atoms, and gas atoms under irradiation conditions, respectively; R vi For the annihilation terms of point vacancies and interstitial atoms; S v S i This is the absorption term of grain boundaries for vacancies and interstitial atoms; K v Vacancy generation rate under cascade-stomatal interaction The expressions for the generation of vacancies and interstitial atoms are as follows: Where R1 and R2 are two randomly generated numbers between 0 and 1; E is a bias constant representing the varying number of vacancies and interstitial atoms generated by the ex-situ damage; parameter P casc V represents the probability of a cascading collision occurring. G This represents the maximum increase in vacancy concentration caused by a single cascade collision event; the condition η < 0.8 ensures that cascade collisions occur only in the matrix phase and not in the porosiform phase. The production rate of gas atoms is: P g =2(1-h) 2 LΩf r Ran (35) Where Λ is a constant; Ω is the lattice volume of the gas atoms; f r It is the fission rate; Ran is a random number between 0 and 1; When vacancies and interstitial atoms meet, they annihilate and recombine to form a perfect crystal lattice, which can be represented by the following formula: R vi =n r c v c i (36) Among them, v r It is the recombination rate: ν r =v b +η 2 v s v b and v s These represent the recombination rates of point defects at the matrix phase and the interface, respectively; v b =4πr iv (D i +D v ) / Ω, r iv It is the composite volume radius; D v and D i These are the diffusion coefficients of vacancies and interstitial atoms, respectively; S v S i The absorption term for vacancies and interstitial atoms at grain boundaries is expressed as follows: in Φ is the grain boundary absorption factor, representing the intensity of absorption of point defects by grain boundaries; Φ = ∑φ i 2 It is a function related to the grain boundary position, where Φ < 1.0 at the grain boundary and Φ = 1.0 inside the grain; The equilibrium concentration of vacancies and interstitial atoms; S5.2 uses the semi-implicit Fourier spectral method to solve the phase field evolution equation, and further simplifies the formula of the phase field evolution equation to obtain: Where f is the sum of the volume free energy density and the polycrystalline interaction energy density; taking a Fourier transform of both sides of the equation yields: In this equation, the variables within parentheses represent the variables after the Fourier transform; k = (k1, k2) are the vector coordinates of the Fourier transform; by implicitly handling linear and second-order operators and explicitly handling other terms, a semi-implicit form of the equation is obtained: Where Δt is the time step between the nth and (n+1)th steps; further simplifying the equation, we get: The evolution equations for vacancy concentration, interstitial atom concentration, and gas atom concentration were solved using a similar method, and the results are shown in the following equation: The equation for the evolution of polycrystalline phase field variables is solved as follows:

7. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S6 includes the following sub-steps: S6.1 The numerical solution results in step S5 are visualized using visualization software to obtain the morphological evolution of pores and grains during the densification of ceramic nuclear fuel irradiation: The visualization variables characterizing the irradiation densification simulation process are: In the formula, For defined visualization variables; inside the grain At the grain boundary Inside the pores Therefore, grains, grain boundaries, and pores can be distinguished by differentiating the values ​​of visualized variables at various points in space; S6.2 uses the visualization variables obtained in S6.1 to write VTK files and perform visualization processing: The VTK file format contains five basic parts: (1). The first part is the file version and identifier; (2). The second part is the title; (3). The third part is the file format. Enter the format name that describes the file type in this part. The format name is either ASCII or BINARY. (4). The fourth part is the structure of the dataset. This part defines the geometric structure of the dataset, including the beginning of the DATASET row and keywords describing the dataset type; (5). The fifth part is the attributes of the dataset. This part begins with the keyword POINT_DATA, followed by the number of specified points; then, enter the actual dataset, i.e., the visualization variables obtained in S6.1; Finally, import the obtained VTK file into Paraview visualization software for visualization processing; S6.3 The porosity is statistically calculated using the simulation results of the phase-field equation. First, the entire simulation region is cyclically processed, and areas where η>0.8 are considered porosity phases and counted. The porosity is then calculated based on the count result N0. Where, N x and N y These represent the number of grid cells in the x and y directions, respectively.

8. The method for determining the effect of the degree of irradiation densification of ceramic nuclear fuel on thermal conductivity as described in claim 1, characterized in that: Step S7 includes the following sub-steps: S7.1 calculates the effective thermal conductivity during the irradiation densification process of ceramic nuclear fuel. Based on the results of solving the phase field evolution equation in S5, the porosity and grain distribution of the nuclear fuel are obtained. To calculate the effective thermal conductivity under the corresponding microstructure, the temperature distribution related to the microstructure is obtained by solving the steady-state heat conduction equation. In the formula, κ(η,φ) is the local thermal conductivity, which is related to the heat transfer performance corresponding to the microstructure. It is assumed that from the inside of the grain, κ(η,φ) = κ bulk κ bulk The thermal conductivity is the thermal conductivity under the condition of a complete crystal lattice structure. The presence of grain boundaries will disrupt the integrity of the crystal lattice arrangement and reduce the heat transfer performance. Therefore, it is necessary to define the thermal conductivity at the grain boundary as κ(η,φ)=κ GB κ GB The thermal conductivity at the grain boundary; when located within the pores, κ(η,φ)=κ bubble κ bubble The thermal conductivity within the pores is given by the following formula: The steady-state heat conduction equation is solved using the finite difference algorithm to obtain the equilibrium temperature field; the effective thermal conductivity is calculated using the following formula: Where j Q It is heat flux, T l and T r These are the average temperatures of the left and right boundaries, respectively. S7.2 Based on statistical porosity, the irradiation densification process is monitored by tracking the shrinkage of all initial pores over time, as defined below: Where Sh(t) is the stomatal shrinkage rate at time t; A(0) and A(t) are the stomatal areas at the initial time and time t, respectively.

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