Method and system for obtaining TRISO type fuel particle failure probability

Abstract

The invention discloses a TRISO type fuel particle failure probability obtaining method and system, and relates to the technical field of nuclear reactor accident safety analysis, and the method comprises the following steps: obtaining the temperature of any position in a TRISO type fuel particle model, the internal pressure generated by the fission gas release amount, and the external pressure; acquiring various strain types of material behaviors of a pyrolytic carbon layer in the TRISO type fuel particle model under an irradiation condition, constructing a strain model, and acquiring radial and tangential stress at any position according to the strain model; the tensile stress of the pyrolytic carbon layer to the SiC layer is determined through the radial or tangential stress at any position, and the failure probability is obtained according to the tensile stress. According to the method, the relieving effect of creepage and swelling of the pyrolytic carbon material caused by the irradiation effect on the tangential tensile stress of the SiC layer is considered, comprehensive mechanical evaluation is carried out on the fuel structure, and the prediction accuracy of the TRISO type fuel particle failure probability is greatly improved.

Description

Technical Field

[0001] The present invention relates to the technical field of nuclear reactor accident safety analysis, and in particular to a method and system for obtaining the failure probability of TRISO type fuel particles. Background Art

[0002] Tristructural isotropic (TRISO) fuel particles can be said to be the strongest form of nuclear fuel ever. TRISO fuels are currently used in many reactor types, including high temperature gas-cooled reactors, light water reactors, micro reactors, nuclear thermal propulsion and molten salt reactors. TRISO fuel particles include OPyC (outer dense pyrolytic carbon layer), SiC (silicon carbide layer), IPyC (inner dense pyrolytic carbon layer), buffer (loose pyrolytic carbon layer) and core from the outside to the inside, and fuel is arranged inside the core. This all-ceramic coated fuel particle is the first barrier to block the release of fission products. Its main function is to provide fission heat and block the release of fission products. Under high temperature and high neutron injection rate operating conditions, gaseous fission products and gases such as CO will be produced in the fuel core of the coated fuel particles. These gaseous fission products are blocked by the four layers of the outer coating and retained inside the coated fuel particles. Therefore, the performance of the coated fuel particles under thermal load, high irradiation flux and complex mechanical conditions determines the safety of the reactor, and it is of great significance to carry out research on it.

[0003] At present, the main mechanisms that cause the damage of coated fuel particles are: manufacturing damage rate, SiC thermal decomposition, amoeba effect, pressure shell damage, and palladium Pd erosion of silicon carbide. SiC thermal decomposition will only occur at high temperature (greater than 2100℃) and high flux, and can generally be ignored. The amoeba effect and Pd erosion of silicon carbide have little effect on the damage of coated fuel particles, so pressure shell damage is the most important mechanism causing fuel particle damage. At present, commercial software such as COMSOL and ABAQUS are mainly used in China to carry out some work on TRISO fuel. However, it cannot be coupled with reactor physics and thermal safety analysis programs, and the calculation efficiency is low. Most of the physical models currently developed (such as the widely used German PANAMA program) assume that TRISO fuel particles are microsphere elastic pressure vessels, and that the SiC layer is the only rigid load-bearing layer. The creep and swelling of pyrolytic carbon materials caused by the irradiation effect do not take into account the relief of the tangential tensile stress of the SiC layer, resulting in an over-prediction of the failure probability of TRISO fuel particles. Summary of the invention

[0004] The present invention provides a method and system for obtaining the failure probability of TRISO type fuel particles, which solves the problem that the prior art does not consider the mitigation effect of creep and swelling of pyrolytic carbon materials caused by radiation effect on the tangential tensile stress of SiC layer, resulting in an over-prediction of the failure probability of TRISO type fuel particles.

[0005] In a first aspect, the present invention provides a method for obtaining a failure probability of a TRISO type fuel particle, comprising the following steps:

[0006] Construct a TRISO fuel particle model, and obtain the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the fission gas release, and the external pressure;

[0007] Obtaining multiple strain types of material behaviors of the pyrolytic carbon layer in the TRISO type fuel particle model under irradiation conditions, and constructing a strain model based on the multiple strain types; obtaining radial and tangential stresses at any position in the TRISO type fuel particle model according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; wherein the multiple strain types include elastic strain, creep strain, swelling strain, and thermal expansion strain;

[0008] The tensile stress of the pyrolytic carbon layer on the SiC layer is determined by the radial or tangential stress at any position, and the failure probability of the TRISO type fuel particle model is obtained according to the tensile stress.

[0009] Preferably, obtaining the temperature at any position in the TRISO type fuel particle model comprises the following steps:

[0010] The TRISO type fuel particle model is divided into n nodes along its radial direction;

[0011] Obtaining the conduction equation in the spherical coordinate system with internal heat source of the TRISO type fuel particle model, and deriving the temperature equation based on the conduction equation;

[0012] Based on the quasi-steady-state assumption, the temperature at any radius of the TRISO fuel particle model is obtained;

[0013] Substituting the temperature at any radius into the temperature equation, we get the heat conduction equation for the multilayer structure:

[0014] Given the outer surface temperature of the TRISO fuel particle model, the temperature at any radius of each layer is obtained in turn through the multi-layer structure heat conduction equation.

[0015] Preferably, obtaining the internal pressure generated by the amount of fission gas released comprises the following steps:

[0016] Derivation of a first release fraction based on a direct recoil release process of fission gas, and derivation of a second release fraction based on a diffusion release process of fission gas;

[0017] obtaining a fission gas release amount by combining the fission gas production amount, the first release fraction, and the second release fraction;

[0018] The fission gas release amount is input into the RK state equation to obtain the internal pressure generated by the fission gas release amount.

[0019] Preferably, the strain model is constructed based on multiple strain types, as shown below:

[0020]

[0021] In the formula, ε r is the radial strain, ε t is the tangential strain, E is the elastic modulus, σ r is the radial stress, σ t is the tangential stress, μ is the Poisson's ratio, c is the creep coefficient, υ is the Poisson's ratio in creep, S r is the irradiation-induced radial dimension change rate, α r is the radial thermal expansion coefficient, t is the time, S t is the irradiation-induced tangential size change rate, α t is the tangential thermal expansion coefficient, is the rate of temperature change.

[0022] Preferably, the step of obtaining the radial and tangential stresses at any position in the TRISO type fuel particle model according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure comprises the following steps:

[0023] Transform the strain model into equations related to radial and tangential strains;

[0024] The equations for radial strain and tangential strain are simplified to obtain general expressions for radial stress and tangential stress;

[0025] Obtaining a radial stress balance equation for each interface in a TRISO type fuel particle model, and obtaining each interface stress according to the radial stress balance equation;

[0026] Each interface stress, the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the fission gas release, and the external pressure are input into the general expression of radial stress and tangential stress to obtain the radial and tangential stresses at any position.

[0027] Preferably, the equations for radial strain and tangential strain are simplified as follows:

[0028]

[0029] in,

[0030]

[0031] In the formula, σ r (r,t) is the radial stress at any position, σ t (r, t) is the tangential stress at any position, u(r, t) is the radial displacement at any position, r a is the inner radius of the layer, r b is the outer radius of the layer, r is the radius of the node, p is the internal pressure, q is the external pressure, F(t) is the physical property function required to calculate the creep and swelling deformation caused by radiation, n is the time increment, and a0 is the coefficient related to the swelling strain rate, thermal expansion and creep.

[0032] Preferably, the failure probability of the TRISO type fuel particle model is obtained according to the tensile stress, as shown below:

[0033]

[0034] Where f(t,T) is the failure probability of the node at time t and temperature T, σ t is the resultant force of internal and external forces, T is the node temperature, σ0 is the strength of the SiC layer after irradiation, and m is the shape parameter of the Weibull distribution of the SiC layer strength.

[0035] In a second aspect, the present invention provides a TRISO type fuel particle failure probability acquisition system, comprising:

[0036] A construction module is used to construct a TRISO type fuel particle model, and obtain the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure;

[0037] An acquisition module is used to acquire multiple strain types of material behaviors of the pyrolytic carbon layer in the TRISO type fuel particle model under irradiation conditions, construct a strain model based on the multiple strain types, and acquire radial and tangential stresses at any position according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; wherein the multiple strain types include elastic strain, creep strain, swelling strain, and thermal expansion strain;

[0038] The calculation module is used to determine the tensile stress of the pyrolytic carbon layer on the SiC layer through the radial or tangential stress at any position, and obtain the failure probability of the TRISO type fuel particle model according to the tensile stress.

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

[0040] The present invention first obtains the basic conditions for calculating the failure probability of TRISO-type fuel particles, namely, the temperature at any position, the internal pressure generated by the amount of fission gas released, and the external pressure. Then, the thermodynamic behavior of the multi-layer spherical shell is analyzed, and various strain types of material behaviors of the pyrolytic carbon layers on both sides of the SiC layer in the TRISO-type fuel particle model under irradiation conditions are obtained, and a strain model is constructed based on multiple strain types. The strain model is combined with the obtained basic conditions to obtain the radial and tangential stresses at any position. The present invention considers factors such as thermal effects, elasticity, creep and thermal expansion under irradiation to analyze the radial and tangential stress conditions at any position in the shell. Finally, the tensile stress of the pyrolytic carbon layer on the SiC layer is determined by the radial or tangential stress at any position, and the failure probability of the TRISO-type fuel particle model is obtained according to the tensile stress. The present invention takes into account the relieving effect of the creep and swelling of the pyrolytic carbon material caused by the irradiation effect on the tangential tensile stress of the SiC layer, and conducts a comprehensive mechanical evaluation of the fuel structure, which solves the problem of incomplete consideration of factors and greatly improves the prediction accuracy of the failure probability of TRISO-type fuel particles. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0042] Figure 1 A flowchart of a method for obtaining the failure probability of TRISO type fuel particles according to the present invention;

[0043] Figure 2 It is a schematic diagram of the thermodynamic structure of the TRISO type fuel particles of the present invention;

[0044] in, Figure 2 (a): Schematic diagram of the three-dimensional structure of TRISO fuel particles. Figure 2 (b): Schematic diagram of the cross section of TRISO type fuel particles;

[0045] Figure 3 It is a dual-parameter Maxwell model diagram of the present invention;

[0046] Figure 4 It is a schematic diagram of the force analysis of the TRISO type fuel particles of the present invention;

[0047] Figure 5This is a comparison of the tangential stress results for Case 8 of the IAEA CRP-6 benchmark. DETAILED DESCRIPTION

[0048] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0049] Reference Figure 1 The present invention provides a method for obtaining the failure probability of TRISO fuel particles, and specifically proposes a mechanism model that can be used to evaluate the thermodynamic properties, irradiation properties and coating shell failure probability of TRISO fuel in a complex coupling field. The model analyzes the heat conduction of a sphere with an internal heat source and the heat exchange of an air gap by a quasi-steady-state method to obtain the overall temperature distribution of the fuel, providing thermal boundary conditions for mechanical analysis. At the same time, the generation of internal gas is analyzed by comprehensively considering the direct recoil effect and the equivalent sphere diffusion model, and the internal pressure is calculated in combination with the RK gas state equation to provide pressure boundary conditions for mechanical solution. For the analysis of the thermodynamic behavior of multilayer spherical shells, based on the viscoelastic dual-parameter Maxwell constitutive model of mechanical stress-strain-displacement of multilayer pressure shells, the closed-form analytical solution of the complex multilayer structure mechanical model is theoretically derived by the simplified evolution of the series solution, and the radial and tangential stress conditions and displacements at any position in the shell are analyzed considering factors such as irradiation, thermal effect, elasticity, creep, and thermal expansion. Finally, the pressure shell damage of TRISO fuel is evaluated by thermal analysis, irradiation analysis, and mechanical analysis.

[0050] The TRISO-type coated fuel studied in this invention is a multi-layered microsphere shell prepared by chemical deposition method CVD, with a diameter of only about 1 mm. It is subjected to the internal fission gas pressure P and the external system environmental pressure q, and bears the core fission heat load. The geometric structure of the coated fuel is divided into Figure 2 As shown, in the analysis, it is divided into n nodes along the radial direction for calculation. The specific steps of the present invention are as follows:

[0051] Step 1: Calculation of heat transfer of multi-layer structures with internal heat sources based on the quasi-steady-state method.

[0052] The present invention aims to provide a novel TRISO type fuel performance analysis code, which can be coupled with a system analysis program for analysis. Therefore, the lumped parameter temperature is given by the system program, and the temperatures of OPyC (outer dense pyrolytic carbon layer), SiC (silicon carbide layer), IPyC (inner dense pyrolytic carbon layer), buffer (loose pyrolytic carbon layer) and core are calculated inwardly through the thermal conductivity of each layer of material to determine the temperature distribution of the entire particle.

[0053] The following assumptions are made for the temperature solutions at the particle centerline and at the interface of each particle layer: 1) The entire TRISO-type fuel particle is spherically symmetrically distributed, with no particle defects or failures; 2) The material of the TRISO-type fuel particle is isotropic; 3) The thermal properties of the TRISO-type fuel particle material depend only on temperature; 4) The gap between the buffer and IPyC, if formed, can be regarded as a heat conductive medium like all media; 5) The contact resistance between the layers within the TRISO-type fuel particle can be ignored; 6) Heat generation inside the TRISO-type fuel particle only occurs in the core.

[0054] The general conduction equation in spherical coordinates with internal heat sources for the TRISO type fuel particle model is as follows:

[0055]

[0056] It can be further concluded that:

[0057]

[0058] Since the temperature calculation of TRISO fuel particles at any radius is based on the quasi-steady state assumption, therefore:

[0059]

[0060] Substituting formula (3) into formula (2) we can further derive:

[0061]

[0062] in, is the volume heat release rate, W / m 3 ; ρ is density, Kg / m 3 ;c p is the specific heat capacity, J / (Kg·K); k is the thermal conductivity, W / (m·K); V k is the core volume, m 3 ;Q k is the core heat flow, W; Q b is the buffer layer heat flow, W; Q gap is the heat flux between the buffer layer and the IPyC layer, W; Q iis the heat flow in the IPyC layer, W; Q s is the heat flow of SiC layer, W; Q o is the heat flow of the OPyC layer, W; is the core volume heat release rate, W / m 3 ; r i is the node radius, m; k is the thermal conductivity of the material, W / (m·K), T is the node temperature, R is the outer radius, r is the node radius, T R is the outer surface temperature.

[0063] When the outer surface temperature of the TRISO fuel particle is given, the temperature T at any radius of each layer of the TRISO fuel particle can be calculated in sequence through the physical properties of each material (UO2 fuel, air gap, coating material). i .

[0064] The initial structure of TRISO fuel particles varies from manufacturer to manufacturer. Under the combined effects of in-pile irradiation and thermodynamics, the geometric changes of the nodes are determined by S5-S8. At the same time, the air gap between buffer and IPyC is determined based on the relative position of the interface, and the S2 air gap heat transfer model is activated. The thermal conductivity of the air gap composed of a mixture of multiple gases is calculated by the second step MATPRO library.

[0065] Step 2: Calculate the gap heat transfer of mixed gas by combining with MATPRO library.

[0066] The correlation used to calculate the gap thermal conductivity is based on the MATPRO library. The gas in TRISO fuel particles is not composed of a single gas, but a mixture of multiple gases. Therefore, it is necessary to determine the thermal conductivity of the mixed gas, and then calculate the air gap temperature through the first step of the heat conduction equation (4). The thermal conductivity of the mixed gas is calculated as follows:

[0067]

[0068] Where:

[0069]

[0070] Where K mix is the thermal conductivity of the mixed gas, W / (m·K); n is the number of gas species in the mixed gas; x i is the mole fraction of the i-th gas; is δ ij is the Kronecker function; M i is the relative molecular mass of the i-th gas; k i is the thermal conductivity of the ith gas, W / (m·K).

[0071] The gases generated in TRISO fuel particles are mainly Xe, Kr, and CO gases. The calculation formula for pure inert gas or diatomic gas is as follows:

[0072] k=AT B (6);

[0073] Where k is thermal conductivity, W / (m·K); T is gas temperature, K; A and B are gas constants, as shown in Table 1:

[0074] Table 1 Gas thermal conductivity related constants

[0075]

[0076]

[0077] Step 3: Calculate the amount of fission gas released by considering the recoil effect and the equivalent spherical diffusion model.

[0078] When the uranium dioxide core undergoes fission reaction, various radioactive nuclides will be produced. The fission gases are mainly Xe and Kr, which are stable and do not react with other substances. The release of fission gas will increase the internal gap pressure, reduce the thermal conductivity of the air gap, and increase the release of fission gas, aggravating the deterioration of fuel performance. Consider that fission gas release (FGR) is released by direct recoil release and diffusion release.

[0079] FGR=(F recoil +[1.0-F recoil ]F Booth )FGP (7);

[0080] Among them, F recoil and F Booth are the release fractions of direct recoil and diffuse fission gases, namely the first release fraction and the second release fraction, respectively, and FGP is the fission gas production in the core.

[0081] For TRISO type fuel, the internal gas includes not only the fission gas Kr / Xe, but also the considerable amount of CO produced by the release of oxygen atoms due to the fission of UO2 and the reaction with carbon in the material.

[0082] For fission gas Kr / Xe, the amount of material produced can be determined by the burnup Bu and the fission yield Γ.

[0083]

[0084] Where V is the volume of the core, m 3 ; N avo is Avogadro's constant, 6.022*10 23 at / mol; Γ FG It is the sum of the fission yields of Kr and Xe atoms each time they fission, which is approximately 0.297.

[0085] There is still no more mechanistic model for the generated CO, so it is generally believed that all the released oxygen combines with the carbon in the buffer layer to form carbon dioxide. The more commonly used empirical model is the empirical relationship given by Proksch et al. to calculate the amount of carbon monoxide generated at each finite element integration point in the fuel:

[0086]

[0087] Where O / F( / ) is the released oxygen atoms, atoms / fission; t is the irradiation time, s; T is the time-averaged surface temperature during the irradiation, K.

[0088] At high irradiation temperatures and long irradiation times, the diffusion processes are fast enough to establish equilibrium in the TRISO type fuel particles. Under these conditions, the dependence on temperature and time disappears and the oxygen released is given by the thermodynamic upper limit. For UO2, the maximum value of O / F is 0.4.

[0089] The direct kinetic release of fission gases from the core to the buffer, i.e. direct recoil, is explained by taking into account geometric factors and the range of fission fragments derived from experimental data. The fission gas mixture consists mainly of Kr and Xe in relative fractions of 18.5% and 81.5%, so the recoil fraction is given by:

[0090] F recoil =0.185F recoil,Kr +0.815F recoil,Xe (10);

[0091] in:

[0092]

[0093] Where OU and CU are the initial oxygen-to-uranium and carbon-to-uranium ratios; ρ is the density of the core, g / cm 3 ; As shown in the following table:

[0094] Table 2 Parameter range

[0095]

[0096] The diffusion release from the core to the grain boundary and the subsequent transport through the interconnected pores are calculated using the Booth equivalent sphere diffusion model. The conservation equation is:

[0097]

[0098] Therefore, the Booth release fractions of short-lived and long-lived fission products can be derived as follows:

[0099]

[0100] Among them, r grain is the radius of the diffusion sphere (i.e., the average grain radius), m; D is the diffusion rate of fission gas in the grain, m 2 / s; D' = D / r grain 2 is the reduced diffusion coefficient, s -1 ; t time, s; τ = D't; μ = λ / D'.

[0101] For the undisturbed diffusion coefficient D, the form given in the PARFUME code for UO2 fuel was used.

[0102] D = D1 + D2 + D3 (14);

[0103] Where D i , i=1, 2, 3 are the diffusion coefficients of the mechanisms controlling diffusion in different temperature ranges.

[0104] At the highest temperatures, diffusion proceeds through the cation lattice via thermally activated vacancies.

[0105]

[0106] Where T k is the temperature of the inner core, K.

[0107] At intermediate temperatures, diffusion is driven by vacancies created during the irradiation process.

[0108]

[0109] in:

[0110]

[0111] Among them, s(m) is the atomic hopping distance; jv(s-1) is the cation vacancy hopping rate; K'(s-1) is the defect generation rate per atom; Z( / ) is the number of sites around the point defect where recombination inevitably occurs; K(104defects / fission) is the damage rate; Bu(FIMA) is the burnup.

[0112] At lower temperatures the fission rate is proportional to the density.

[0113] D3=2.0×10 -40 f ″′ (18).

[0114] Step 4: Determine the internal pressure using the Redlich-Kwong equation of state.

[0115] The third step can determine the current amount of fission gas, namely FGR, and then the gas state equation can be used to calculate the pressure under the current volume and amount of gas. Since the structure of the gas stored inside the TRISO fuel particles is a loose pyrolytic carbon layer buffer, the Redlich-Kwong state equation, which is considered to be the most suitable for porous materials and mixed gases, is used when calculating the gas pressure inside the particles to solve the displacement and stress for subsequent calculations. That is, the internal pressure P:

[0116]

[0117] Among them, V g,i is the molar volume of gas i, equal to V buffer体积 / FGR;a i and b i is the RK gas constant, as shown below:

[0118]

[0119] Through the above steps, the internal pressure boundary p of the coating layer and the temperature T of each node are calculated. i , combined with the external environmental pressure boundary q and the temperature-related physical parameters, the stress on the RTISO fuel coating can be derived and solved.

[0120] Step 5: Determine the mechanical constitutive relationship based on the two-parameter Maxwell creep model of viscoelastic theory.

[0121] Through S1-S4, the temperature distribution of TRISO fuel particles and the internal pressure boundary of the fission gas released inside can be calculated and determined, combined with the node temperature T i And pressure boundaries p, q, the stress analysis of the coating material can be carried out by deriving the mechanical constitutive relationship of the multilayer pyrolytic carbon material.

[0122] The material behavior of the pyrolytic carbon layer is represented by the two-parameter Maxwell creep model for radiation-induced creep, which appropriately represents the steady-state or secondary creep exhibited by the pyrolytic carbon material and can be used to derive the mechanical model of the multilayer pressure-bearing coating. This model can be Figure 3 The total strain rate obtained by adding the strain rates of the spring and the buffer is:

[0123]

[0124] Taking into account the Poisson effect and the radial and tangential stress strains in the spherical geometry of TRISO-type fuel particles, and including the strains due to expansion, equation (21) is replaced by the following two equations:

[0125]

[0126] Where, ε is radial or tangential strain; σ is radial or tangential stress; E is elastic modulus, MPa; μ is Poisson's ratio; c is creep coefficient, 10 21 cm 2 / Pa;υ is Poisson's ratio in creep; S r and S t is the rate of radial or tangential dimensional change induced by irradiation.

[0127] The four terms on the right side of (22) and (23) represent the strain rate of the pyrolytic carbon layer modeled in this model: the first term represents the elastic strain rate caused by the radial and tangential stress components, The second term represents the irradiation-induced creep strain rate caused by the stress component, cσ; the third term represents the irradiation-induced swelling strain rate, S; and the fourth term represents the strain rate caused by thermal expansion,

[0128] Step 6: Derive the closed-form analytical solutions of the mechanical constitutive equations (22) and (23) from the simplified evolution of the series solution.

[0129] Due to the strain adaptation of the anisotropic thermal expansion to the temperature changes that may occur during irradiation, these equations include secondary creep (creep strain rate is proportional to stress) that is characteristic of pyrolytic carbon materials. The following strain-displacement relationship and equilibrium requirements for the spherical system describe the behavior of the pyrolytic carbon layer:

[0130]

[0131] The above equations also describe the behavior of the SiC layer except that the creep and expansion terms are omitted. If the expansion or thermal expansion in the pyrolytic carbon layer is isotropic, the radial strain component and the tangential strain component can be set equal.

[0132] In order to r , tangential stress σ t and the node displacement u, assuming that the solution of the above equations (22)-(24) is in the form of the following very general series solution:

[0133]

[0134] where i is the number of terms and is the index at time t; the t0 term is included in these sums to accommodate the presence of internal or external pressure at time zero. Substituting this solution into equations (22)-(24), the general equation for displacement is obtained:

[0135]

[0136] In these expressions, the swelling strain rate and thermal expansion strain rate functions are expanded into series, such as Sr =Σ(S r ) i t i The function F(t) is defined as follows:

[0137]

[0138] This function includes the physical properties required to determine the stress affected by coating creep c, swelling S and thermal expansion deformation α, which are the last three terms on the right side of equations (22) and (23). Substituting equation (27) into (28) and taking the derivative with respect to t, the equation of the function F(t) is:

[0139]

[0140] The overline in the formula represents the numerical average of the expansion strain rate and thermal expansion strain rate with time increment, and is regarded as a constant through the increment. The general solution of this differential equation is a closed function:

[0141]

[0142] Among them, a0 of time increment n is:

[0143]

[0144] By solving the purely elastic problem σ r0 and σ t0 , indicating that the quantity f0 is zero, because the radiation has no effect at time zero. Equation (26) can be used to solve for the displacement term u i , and then combined with the equilibrium equation, we can express σ ri and σ ti .

[0145]

[0146] In order to solve the unknown quantity A in the formula i , B i , expressing the stresses p(t) and q(t) acting on the inner and outer surfaces as time series terms (35), and using equation (33) to equate the radial stress with these pressures on the surface, yields the following coefficient relationship.

[0147]

[0148] The above equations, combined with equations (32)-(34), can be used to develop a closed-form analytical solution that is derived from a simplification of the series solution and solves for the radial displacement at any radial location in the spherical shell that exhibits expansion and creep in addition to normal elastic behavior. Substituting equations (36)-(37) into equation (33) and the result into equation (25), adding all the terms in the series, and combining them with equation (37) yields the following radial stress σ in the spherical shell: r (r, t), tangential stress σ t The general expression for the radial displacement u(r,t) is:

[0149]

[0150] Among them, r a and r b are the inner and outer radii of the layer, respectively, and the radial stresses (or pressures) p and q acting on the inner and outer surfaces of the layer are considered to be positive outward. i Depends on the geometry and properties of the layer and the radius r.

[0151] The general expressions of radial stress, tangential stress and radial displacement in the shell show that if the internal and external stresses / pressures of the layer, i.e., p / q, and the radial stress equilibrium equation at the interface are determined, the stresses (σ t / σ r ) and displacement (u).

[0152] Step 7: Solve the radial stress equilibrium equation at the interface.

[0153] In TRISO fuels, in order to determine the stress and displacement at any node, it is necessary to determine the radial stress σ at the IPyC / SiC and SiC / OPyC interfaces. rO / σ rI , so that the general expressions of radial stress and tangential stress can be obtained. The radial stress at the interface is solved by solving the displacement equation at the interface and the differential equation for t, and the two simultaneous differential equations are as follows:

[0154]

[0155] Among them B i The expression is as follows:

[0156]

[0157] Z = b2c1-(c2-d1)(b1-a2)

[0158] The expressions of the functions x(t) and y(t) in terms of time increments are as follows:

[0159]

[0160] Combining the above equations, we can get the solution of equations (41) and (42):

[0161]

[0162] where x0, x1, y0, y1, m1, m2, υ0, υ1, w0, and w1 are constant during a time increment but change from one increment to the next. i , b i 、c i ,d i Depends on K solved for according to the geometry and properties of the layer and the radius r i .

[0163] The solution is applied in time increments, which means that the coefficients D1 and D2 for each increment are determined from the initial conditions of that increment. At the beginning of irradiation (t = 0), the initial internal pressure p0 and external pressure q0 are applied, and the integral terms in the displacement general solution all disappear. After solving, the radial interface stress σ at the beginning of irradiation is obtained rO (0) / σ rI (0) is as follows:

[0164]

[0165] These become the initial conditions for determining the coefficients D1 and D2 for the first increment. In subsequent time increments, the radial stress at the end of one increment becomes the initial condition for the next increment. Using equation (44), the coefficients for a general time increment n are determined as:

[0166]

[0167] in:

[0168]

[0169] And t n-1 represents the time t at the end of the previous increment n-1. In applying these equations, all material properties, dilatation strain rates, thermal expansion strain rates, and known internal and external pressures are numerically averaged over the time increments.

[0170] Step 8: Determine the stress condition at any node by solving equations (38)-(39) in combination with the interface stress.

[0171] Reference Figure 4 , p and q are the forces generated at the interface by the internal pressure of the fission gas and the external pressure, respectively, σ rI and σ tOThese are the forces exerted by IPyC and OPyC on the SiC layer, respectively.

[0172] When the radial contact stress on the inner and outer surfaces of the layer is solved and added to F(t) in equation (30), the radial σ at any radial position in the coating can be determined: r Or tangential stress σ t .

[0173] Step 9: Pressure shell damage model to evaluate fuel cladding integrity.

[0174] Generally speaking, the failure mechanism of TRISO fuel is a function of temperature, fuel consumption, flux, and particle design details. Studies have shown that pressure shell damage of the cladding layer caused by tangential force exceeding the allowable stress range is currently the most important failure form of TRISO fuel. Stress-induced failure of SiC ceramics is a brittle failure. Under tensile stress, material cracks may rapidly expand and cause rapid and complete failure. The uncertainty of material parameters at the time of failure is very large. Weibull distribution has been widely used in modeling the failure probability of ceramic materials. This distribution can be used to describe the failure probability of CVD micro-element under the stress state. Its mathematical expression is as follows:

[0175]

[0176] Where σ0 is the strength of SiC layer after irradiation; σ t is the combined force of internal and external forces; m is the shape parameter of Weibull distribution of SiC layer strength.

[0177] Fast neutron irradiation will cause a decrease in the strength and Weibull modulus of the SiC layer. The strength and modulus of the SiC layer after irradiation can be expressed by (49)-(51).

[0178]

[0179] Among them, σ 00 is the strength of the unirradiated SiC layer, 834 MPa; Φ is the fast neutron flux 10 25 m -2 ; Φ σs Dependent on temperature; the lower limit of σ0 is 196MPa.

[0180]

[0181] H=0.44+0.56e -ηt ;

[0182] η=0.565e -187400 / RT ;

[0183] Where: m0 is the Weibull modulus without irradiation, 8.02; Φ msis an empirical constant related to modulus and temperature; H is the influencing factor of intergranular corrosion, which is 1 when not considered; the lower limit of m is 2.0.

[0184] Example

[0185] In order to verify the correctness and effectiveness of the mechanical properties and failure assessment method of TRISO type fuel based on viscoelastic mechanics theory, a Coordinated Research Projects (CRP) conducted by the International Atomic Energy Agency (IAEA) was used for verification.

[0186] The key part of IAEA CRP-6 includes the benchmark calculation of the fuel performance model under normal HTGR operating conditions. The present invention selects Case 8 in the benchmark problem as a comparative verification. This is a TRISO type fuel particle undergoing the fuel cycle temperature history characteristics in a pebble bed reactor. Assume that the particle undergoes ten cycles, and the temperature of the fuel in each cycle is initially 873K, increases linearly to 1273K, and then immediately decreases back to 873K. The TRISO type fuel is at ambient pressure, and the internal pressure and temperature undergo ten cycles. Each cycle is one tenth of the total irradiation time, that is, 100 days.

[0187] The above conditions were input into the analysis model as boundaries, and the calculated results were compared with the benchmark test results. The corresponding simulation results are shown in Figure 5 As shown in the simulation comparison results, it can be seen that the TRISO type fuel mechanical properties and failure assessment method based on viscoelastic mechanics theory developed by the present invention can effectively predict the mechanical behavior of the coated fuel.

[0188] In summary, the present invention is aimed at TRISO coated fuel, which is currently widely used in new reactors, and the behavior of the fuel under neutron irradiation-thermomechanical coupling is theoretically derived and developed by the numerical model. The overall temperature distribution is obtained by analyzing the heat conduction and air gap heat exchange of the sphere with internal heat source by the quasi-steady-state method, and then the direct recoil effect and the equivalent sphere diffusion model are comprehensively considered and combined with the RK gas state equation to evaluate the internal pressure. For the thermodynamic behavior of the external spherical shell, based on the viscoelastic dual-parameter Maxwell constitutive model of mechanical stress-strain-displacement of the multi-layer pressure shell, the closed-form analytical solution of the complex mechanical model of the multi-layer pressure layer structure is theoretically derived by the simplified evolution of the series solution, and the radial and tangential stress and displacement at any position in the shell are analyzed by considering factors such as irradiation, thermal effect, elasticity, creep, and thermal expansion. Finally, the pressure shell damage of TRISO fuel is evaluated by the results of mechanical analysis. Therefore, the present invention can carry out in-depth research on the thermodynamic behavior and failure mechanism of TRISO fuel by considering the complex coupling field of thermal effect, irradiation effect, and mechanical effect.

[0189] Furthermore, a mechanical analysis of the TRISO fuel structure was carried out based on the two-parameter Maxwell model of viscoelastic theory, and a closed-form solution of stress-strain-displacement applicable to single-layer to multi-layer spherical shell structures was derived. This solution can correctly reflect the mechanical effects between different material layers in the pressure-bearing structure, and makes up for the defect that the existing analysis program simplifies the assumption of a single-layer structure and cannot consider the physical effects between layers.

[0190] Furthermore, considering the kinetic recoil and diffusion conservation of gas atoms, the main gases produced in TRISO fuel using UO2 as fuel are included in the internal pressure boundary analysis, and the correlation of interstitial thermal conductivity under various gas mixture composition conditions based on the MATPRO library is included in the thermal boundary analysis, making the boundary conditions of temperature and pressure, which are the premise of mechanical calculation, more in line with the actual situation.

[0191] Furthermore, the elasticity, thermal expansion, radiation creep, swelling and other behaviors of TRISO-type fuel under the radiation-heat-mechanical coupling were comprehensively considered, and then a comprehensive mechanical evaluation of the fuel structure was carried out, which solved the problem of treating it as a rigid material and not considering all factors, making the evaluation results credible.

[0192] Furthermore, the established mechanism model data interface is relatively complete, the parameters are easy to adjust, and it can be embedded in the commonly used reactor accident analysis program for coupling calculation, which has strong adaptability.

[0193] Based on the same concept, the present invention also provides a TRISO type fuel particle failure probability acquisition system, including a construction module, an acquisition module and a calculation module.

[0194] The construction module is used to construct a TRISO type fuel particle model, and obtain the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure.

[0195] The acquisition module is used to obtain various strain types of material behavior of the pyrolytic carbon layer in the TRISO type fuel particle model under irradiation conditions, and to construct a strain model based on the various strain types; radial and tangential stresses at any position in the TRISO type fuel particle model are obtained according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; the various strain types include elastic strain, creep strain, swelling strain, and thermal expansion strain.

[0196] The calculation module is used to determine the tensile stress of the pyrolytic carbon layer on the SiC layer through the radial or tangential stress at any position, and obtain the failure probability of the TRISO type fuel particle model according to the tensile stress.

[0197] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0198] Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for obtaining the failure probability of TRISO fuel particles, characterized in that: The following steps are involved: Construct a TRISO fuel particle model, and obtain the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; Obtaining multiple strain types of material behaviors of the pyrolytic carbon layer in the TRISO type fuel particle model under irradiation conditions, and constructing a strain model based on the multiple strain types; obtaining radial and tangential stresses at any position in the TRISO type fuel particle model according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; wherein the multiple strain types include elastic strain, creep strain, swelling strain, and thermal expansion strain; The tensile stress of the pyrolytic carbon layer on the SiC layer is determined by the radial or tangential stress at any position, and the failure probability of the TRISO type fuel particle model is obtained according to the tensile stress.

2. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 1, characterized in that: The method of obtaining the temperature at any position in the TRISO type fuel particle model comprises the following steps: The TRISO type fuel particle model is divided into n nodes along its radial direction; Obtaining the conduction equation in the spherical coordinate system with the internal heat source of the TRISO type fuel particle model, and deriving the temperature equation based on the conduction equation; Based on the quasi-steady-state assumption, the temperature at any radius of the TRISO fuel particle model is obtained; Substituting the temperature at any radius into the temperature equation, we get the heat conduction equation for the multilayer structure: Given the outer surface temperature of the TRISO fuel particle model, the temperature at any radius of each layer is obtained in turn through the multi-layer structure heat conduction equation.

3. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 1, characterized in that: The method of obtaining the internal pressure generated by the amount of fission gas released comprises the following steps: Derivation of a first release fraction based on a direct recoil release process of fission gas, and derivation of a second release fraction based on a diffusion release process of fission gas; obtaining a fission gas release amount by combining the fission gas production amount, the first release fraction, and the second release fraction; The fission gas release amount is input into the RK state equation to obtain the internal pressure generated by the fission gas release amount.

4. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 1, characterized in that: The strain model is constructed based on multiple strain types, as shown below: In the formula, ε r is the radial strain, ε t is the tangential strain, E is the elastic modulus, σ r is the radial stress, σ t is the tangential stress, μ is the Poisson's ratio, c is the creep coefficient, υ is the Poisson's ratio in creep, S r is the radial dimension change rate induced by irradiation, α r is the radial thermal expansion coefficient, t is the time, S t is the irradiation-induced tangential size change rate, α t is the tangential thermal expansion coefficient, is the rate of temperature change.

5. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 4, characterized in that: The step of obtaining the radial and tangential stress at any position in the TRISO type fuel particle model according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the amount of fission gas released, and the external pressure comprises the following steps: Transform the strain model into equations related to radial and tangential strains; The equations for radial strain and tangential strain are simplified to obtain general expressions for radial stress and tangential stress; Obtaining a radial stress balance equation for each interface in a TRISO type fuel particle model, and obtaining each interface stress according to the radial stress balance equation; Each interface stress, the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the fission gas release, and the external pressure are input into the general expression of radial stress and tangential stress to obtain the radial and tangential stresses at any position.

6. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 5, characterized in that: The equations for radial strain and tangential strain are simplified as follows: in, In the formula, σ r (r,t) is the radial stress at any position, σ t (r, t) is the tangential stress at any position, u(r, t) is the radial displacement at any position, r a is the inner radius of the layer, r b is the outer radius of the layer, r is the radius of the node, p is the internal pressure, q is the external pressure, F(t) is the physical property function required to calculate the creep and swelling deformation caused by radiation, n is the time increment, and a0 is the coefficient related to the swelling strain rate, thermal expansion and creep.

7. A method for obtaining the failure probability of TRISO type fuel particles as claimed in claim 6, characterized in that: The failure probability of the TRISO fuel particle model is obtained according to the tensile stress, as shown below: Where f(t,T) is the failure probability of the node at time t and temperature T, σ t is the resultant force of internal and external forces, T is the node temperature, σ0 is the strength of the SiC layer after irradiation, and m is the shape parameter of the Weibull distribution of the SiC layer strength.

8. A TRISO type fuel particle failure probability acquisition system, characterized in that: include: A construction module is used to construct a TRISO type fuel particle model, and obtain the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the fission gas release, and the external pressure; An acquisition module is used to acquire multiple strain types of material behaviors of the pyrolytic carbon layer in the TRISO type fuel particle model under irradiation conditions, and to construct a strain model based on the multiple strain types; radial and tangential stresses at any position in the TRISO type fuel particle model are acquired according to the strain model, the temperature at any position in the TRISO type fuel particle model, the internal pressure generated by the amount of fission gas released, and the external pressure; wherein the multiple strain types include elastic strain, creep strain, swelling strain, and thermal expansion strain; The calculation module is used to determine the tensile stress of the pyrolytic carbon layer on the SiC layer through the radial or tangential stress at any position, and obtain the failure probability of the TRISO type fuel particle model according to the tensile stress.

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