A method and system for obtaining failure probability of TRISO fuel particles

Abstract

The present invention discloses a method and system for obtaining the failure probability of TRISO-type fuel particles, relating to the technical field of nuclear reactor accident safety analysis. The method comprises the following steps: obtaining the temperature, internal pressure generated by the amount of fission gas released, and external pressure at any location in a TRISO-type fuel particle model; obtaining multiple strain types of the material behavior of the pyrolytic carbon layer in the TRISO-type fuel particle model under irradiation conditions, constructing a strain model, and obtaining radial and tangential stresses at any location based on the strain model; determining the tensile stress of the pyrolytic carbon layer on the SiC layer using the radial or tangential stress at any location, and obtaining the failure probability based on the tensile stress. The present invention considers the effect of creep and swelling of the pyrolytic carbon material caused by irradiation on the tangential tensile stress of the SiC layer, performs a comprehensive mechanical assessment of the fuel structure, and significantly improves the accuracy of predicting the failure probability of TRISO-type fuel particles.

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] Tristructurally isotropic (TRISO) fuel particles are arguably the strongest form of nuclear fuel ever developed. TRISO fuel is currently used in many reactor types, including high-temperature gas-cooled reactors (HTGRs), light-water reactors (LWRs), micro-reactors (MNSRs), nuclear thermal propulsion, and molten salt reactors. From the outside in, a TRISO fuel particle consists of an outer dense pyrolytic carbon layer (OPyC), a SiC (silicon carbide) layer, an inner dense pyrolytic carbon layer (IPyC), a buffer layer (loose pyrolytic carbon layer), and a core, with fuel contained within the core. This all-ceramic coated fuel particle serves as the first barrier to the release of fission products, its primary function being to provide fission heat and prevent the release of fission products. Under high-temperature and high-neutron flux operating conditions, gaseous fission products and gases such as CO are generated within the fuel core of the coated fuel particle. These gaseous fission products are trapped within the coated fuel particle by the four outer cladding layers. Therefore, the performance of coated fuel particles under thermal loads, high irradiation fluxes, and complex mechanical conditions determines reactor safety, making their research of paramount importance.

[0003] Currently, the main mechanisms contributing to the failure of coated fuel particles include manufacturing failure rate, SiC thermal decomposition, the amoeba effect, pressure shell failure, and palladium (Pd) corrosion of silicon carbide. SiC thermal decomposition occurs only at high temperatures (greater than 2100°C) and high flux, and can generally be ignored. The amoeba effect and Pd corrosion of SiC have minimal impact on coated fuel particle failure, making pressure shell failure the most important mechanism. Currently, some work on TRISO fuel has been conducted in China using commercial software such as COMSOL and ABAQUS. However, these models lack coupling with reactor physics and thermal safety analysis programs and exhibit low computational efficiency. Most currently developed physical models (such as the widely used German PANAMA program) assume that TRISO fuel particles are microspherical elastic pressure vessels, assuming that the SiC layer is the sole rigid load-bearing layer. These models fail to consider the role of irradiation-induced creep and swelling of the pyrolytic carbon material in alleviating the tangential tensile stress in the SiC layer, leading to overestimation 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 existing technology does not consider the effect of the creep and swelling of the pyrolytic carbon material caused by the radiation effect on the relief of the tangential tensile stress of the 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 fuel particle, comprising the following steps:

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

[0007] Acquire multiple strain types of material behavior of the pyrolytic carbon layer in a TRISO-type fuel particle model under irradiation conditions, and construct a strain model based on the multiple strain types; acquire radial and tangential stresses at any position in the TRISO-type fuel particle model based on 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;

[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 fuel particle model is divided into n nodes along its radial direction;

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

[0012] Obtain the temperature at any radius of the TRISO fuel particle model based on the quasi-steady-state assumption;

[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 sequence 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] Where, 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 size change rate, α r is the radial thermal expansion coefficient, t is the time, S t is the tangential size change rate induced by irradiation, α t is the tangential thermal expansion coefficient, is the temperature change rate.

[0022] Preferably, obtaining the radial and tangential stresses at any position in the TRISO type fuel particle model based on 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:

[0023] Convert 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 stresses at each interface according to the radial stress balance equation;

[0026] The interfacial stresses, 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 expressions 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] Where, 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 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 fuel particle failure probability acquisition system, comprising:

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

[0037] An acquisition module is used to acquire multiple strain types of material behavior 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 based on 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; 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 based on 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, for the thermodynamic behavior analysis of the multi-layer spherical shell, the various strain types of material behavior 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 the various 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 based on the tensile stress. The present invention takes into account the mitigation 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, solving the problem of incomplete consideration of factors and greatly improving 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 following briefly introduces the drawings required for use in the embodiments or the description of the prior art. 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 any creative work.

[0042] Figure 1 This is a flow chart of a method for obtaining the failure probability of TRISO fuel particles according to the present invention;

[0043] Figure 2 Schematic diagram of the thermodynamic structure of the TRISO 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 cross-section of TRISO fuel particles;

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

[0046] Figure 4 Schematic diagram of force analysis of TRISO 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 problem. DETAILED DESCRIPTION

[0048] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. 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 making creative efforts 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. Specifically, it proposes a mechanistic model that can be used to evaluate the thermodynamic properties, radiation performance, and shell failure probability of TRISO fuel in a complex coupled field. This model uses a quasi-steady-state method to analyze the heat conduction and air gap heat transfer of a sphere with an internal heat source to determine the overall fuel temperature distribution, providing thermal boundary conditions for mechanical analysis. Internal gas generation is analyzed by comprehensively considering direct recoil effects and the equivalent sphere diffusion model, and the internal pressure is calculated using the RK gas equation of state, providing pressure boundary conditions for mechanical solutions. For the analysis of the thermodynamic behavior of multilayer spherical shells, a closed-form analytical solution to the complex multilayer structural mechanical model is theoretically derived through the simplified evolution of series solutions based on a viscoelastic two-parameter Maxwell constitutive model of mechanical stress-strain-displacement for multilayer pressure shells. Radial and tangential stresses and displacements at any location within the shell are analyzed, taking into account factors such as radiation, thermal effects, elasticity, creep, and thermal expansion. Finally, thermal, radiation, and mechanical analyses are used to assess the pressure shell failure of the TRISO fuel.

[0050] The TRISO-type coated fuel studied in this invention is a multi-layered microsphere shell made by chemical deposition method CVD, with a diameter of only about 1mm. 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 that can be coupled with a system analysis program for analysis. Therefore, the system program gives a lumped parameter temperature, and calculates the temperature of OPyC (outer dense pyrolytic carbon layer), SiC (silicon carbide layer), IPyC (inner dense pyrolytic carbon layer), buffer (loose pyrolytic carbon layer) and core inward through the thermal conductivity of each layer of material to determine the temperature distribution of the entire particle.

[0053] The temperature solutions at the particle centerline and at the interface of each particle layer have the following assumptions: 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 conducting 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 occurs only in the core.

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

[0055]

[0056] It can be further concluded that:

[0057]

[0058] Since the temperature calculation at any radius of TRISO fuel particles 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 heat flow in the buffer layer, W; Q gap is the heat flux in the gas gap between the buffer layer and the IPyC layer, W; Q iis the heat flow of IPyC layer, W; Q s is the heat flow of SiC layer, W; Q o is the heat flow in 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 external 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 among manufacturers. S5-S8 determine the geometric changes in the nodes under the combined effects of in-pile irradiation and thermodynamics. Simultaneously, the presence of an air gap between the buffer and the 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 multi-gas mixture is calculated using the second step of the 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 inside 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 i-th gas, W / (m·K).

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

[0072] k=AT B (6);

[0073] Where k is the thermal conductivity, W / (m·K); T is the 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 a uranium dioxide core undergoes fission reactions, various radioactive nuclides are produced. Fission gases, primarily Xe and Kr, are stable and unreactive. Release of these gases increases internal interstitial pressure, reduces interstitial thermal conductivity, and thus increases fission gas release, exacerbating fuel performance degradation. Fission gas release (FGR) is considered to occur through both direct recoil and diffusion.

[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 fuel, the internal gas includes not only the fission gas Kr / Xe, but also the non-negligible 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 currently no mechanistic model for the CO generated, so it is generally assumed that all released oxygen combines with carbon in the buffer layer to form carbon dioxide. The most 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; and T is the time-averaged surface temperature during the irradiation period, K.

[0088] At high irradiation temperatures and long irradiation times, the diffusion process is fast enough to establish equilibrium within the TRISO fuel particles. Under these conditions, the temperature and time dependence disappears, and the released oxygen is given by a thermodynamic upper limit. For UO2, the maximum O / F ratio 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 for UO2 fuel in the PARFUME code was used.

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

[0103] Among them 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 irradiation.

[0108]

[0109] in:

[0110]

[0111] where 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 a point defect where recombination inevitably occurs; K(104 defects / fission) is the damage rate; and Bu(FIMA) is the burnup.

[0112] At lower temperatures the fission rate is proportional to 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 determines the current amount of fission gas, or FGR. The gas state equation can then be used to calculate the pressure at the current volume and gas mass. Because the gas stored inside TRISO fuel particles is a loose pyrolytic carbon buffer, the Redlich-Kwong state equation, considered most suitable for porous materials and mixed gases, is used to calculate the internal gas pressure. This equation is used to solve for displacement and stress in subsequent calculations. This 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] The internal pressure boundary p of the coating layer and the temperature T of each node are calculated through the above steps. 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] The temperature distribution of TRISO fuel particles and the internal pressure boundary of the fission gas released inside can be calculated by S1-S4, combined with the node temperature T i and pressure boundaries p, q, the mechanical constitutive relationship of the multilayer pyrolytic carbon material can be deduced to analyze the stress of the coating material.

[0122] The material behavior of the pyrolytic carbon layer is represented by the two-parameter Maxwell creep model of 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 multi-layer pressure-bearing coating. 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 stresses and 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 the radial or tangential strain; σ is the radial or tangential stress; E is the elastic modulus, MPa; μ is the Poisson's ratio; c is the 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 Equations (22) and (23) represent the strain rates 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; 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 anisotropic thermal expansion to possible temperature changes during irradiation, these equations include secondary creep (creep strain rate proportional to stress), which 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 and tangential strain components 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 increments, and is considered a constant through the increments. The general solution of this differential equation is a closed function:

[0141]

[0142] Where a0 for 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 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 evolves 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), summing all the terms in the series, and combining them with equation (37), yield the following radial stress σ in the spherical shell: r (r, t), tangential stress σ t The general expressions for (r, t) and radial displacement u(r, t) are:

[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 stress / pressure of the layer, i.e. p / q, and the radial stress equilibrium equation at the interface are determined, the stress (σ) of all nodes in the layer can be solved. 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 the displacement equation at the interface and the differential equation for t, resulting in two simultaneous differential equations as shown below:

[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 solutions 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, which is solved based on the geometry and properties of the layer and the radius r i .

[0163] The solution is applied in a time-incremental manner, 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 time 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:

[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, dilatational strain rates, thermal expansion strain rates, and known internal and external pressures are numerically averaged over the time increment.

[0170] Step 8: Determine the stress condition of 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 and external pressures of the fission gas, respectively, σ rI and σ tOThese are the forces exerted by IPyC and OPyC on the SiC layer, respectively.

[0172] When the radial contact stresses on the inner and outer surfaces of the layer are 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, burnup, flux, and particle design details. Studies have shown that pressure shell failure of the cladding layer caused by tangential forces exceeding the allowable stress range is currently the most common failure mode of TRISO fuel. Stress-induced failure of SiC ceramics is brittle failure. Under tensile stress, material cracks may rapidly expand, causing rapid and complete failure. The uncertainty of material parameters at the time of failure is very large. The 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-elements under the stress state they are subjected to. Its mathematical expression is as follows:

[0175]

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

[0177] Fast neutron irradiation will cause the strength and Weibull modulus of the SiC layer to decrease. 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 the 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 fuel based on viscoelastic mechanics theory, a Coordinated Research Project (CRP) conducted by the International Atomic Energy Agency (IAEA) was used for verification.

[0186] A key component of the IAEA CRP-6 includes benchmark calculations of fuel performance models under normal HTGR operating conditions. This paper uses Case 8 from this benchmark as a comparative validation. This involves TRISO fuel pellets undergoing a fuel cycle temperature history in a pebble bed reactor. The pellets are assumed to undergo ten cycles, with the fuel temperature initially increasing from 873 K to 1273 K and then immediately decreasing back to 873 K in each cycle. The TRISO fuel is maintained at ambient pressure, with the internal pressure and temperature cycling over ten cycles. Each cycle is one-tenth of the total irradiation time, or 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 2. Figure 5 From the simulation comparison results, it can be seen that the TRISO fuel mechanical properties and failure assessment method based on viscoelastic mechanics theory developed in this invention can effectively predict the mechanical behavior of the coated fuel.

[0188] In summary, this paper theoretically deduces and develops a numerical model for the behavior of TRISO-clad fuel, currently widely used in new reactors, under neutron irradiation and thermomechanical coupling. Using a quasi-steady-state approach, the overall temperature distribution is analyzed for heat conduction and air gap heat transfer within a sphere with an internal heat source. Furthermore, the internal pressure is assessed by comprehensively considering direct recoil effects and the equivalent sphere diffusion model, combined with the RK gas equation of state. Regarding the thermodynamic behavior of the external spherical shell, a closed-form analytical solution for the complex mechanical model of the multilayer pressure shell structure is theoretically derived via a simplified evolution of series solutions based on a viscoelastic two-parameter Maxwell constitutive model of mechanical stress-strain-displacement for the multilayer pressure shell. Radial and tangential stresses and displacements at any location within the shell are analyzed, taking into account factors such as irradiation, thermal effects, elasticity, creep, and thermal expansion. Finally, the results of the mechanical analysis are used to assess the pressure shell failure of TRISO-type fuel. Therefore, by considering the complex coupled field of thermal, irradiation, and mechanical effects, this paper enables in-depth research on the thermodynamic behavior and failure mechanisms of TRISO-type fuel.

[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 stress-strain-displacement solution applicable to spherical shell structures from single-layer to multi-layer was derived. This solution can correctly reflect the mechanical effects between different material layers in the pressure-bearing structure, making 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, by considering the kinetic recoil and diffusion conservation of gas atoms, the main gases generated in TRISO-type fuels fueled by UO2 are incorporated into the internal pressure boundary analysis. At the same time, the correlation of interstitial thermal conductivity under various gas mixture composition conditions based on the MATPRO library is incorporated into the thermal boundary analysis. This makes the temperature and pressure boundary conditions, which are the prerequisites for mechanical calculations, more realistic.

[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 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 true and credible.

[0192] Furthermore, the established mechanism model data interface is relatively complete, the parameters are easy to adjust, and it can be embedded in commonly used reactor accident analysis programs for coupled calculations, with strong adaptability.

[0193] Based on the same concept, the present invention also provides a TRISO type fuel particle failure probability acquisition system, which includes 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 release of fission gas, and the external pressure.

[0195] The acquisition module is used to obtain multiple strain types of material behavior of the pyrolytic carbon layer in the TRISO-type fuel particle model under irradiation conditions, and construct a strain model based on the multiple strain types; the 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 release of fission gas, and the external pressure; the multiple 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 additional 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 modifications and variations to the present invention without departing from the spirit and scope of the present invention. Thus, if such modifications and variations fall within the scope of the claims and their equivalents, the present invention is intended to include such 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, internal pressure generated by the fission gas release, and external pressure at any position in the TRISO fuel particle model; Acquire multiple strain types of material behavior of the pyrolytic carbon layer in a TRISO-type fuel particle model under irradiation conditions, and construct a strain model based on the multiple strain types; acquire radial and tangential stresses at any position in the TRISO-type fuel particle model based on 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 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 based on the tensile stress. The strain model is constructed based on multiple strain types, as shown below: ； ； Where, 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 Poisson's ratio, c is the creep coefficient, υ is the Poisson's ratio in creep, S r is the irradiation-induced radial size change rate, is the radial thermal expansion coefficient, t For time, S t is the irradiation-induced tangential size change rate, is the tangential thermal expansion coefficient, is the temperature change rate; The step of obtaining radial and tangential stresses at any position in the TRISO fuel particle model based on the strain model, the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the amount of fission gas released, and the external pressure comprises the following steps: Convert 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 stresses at each interface according to the radial stress balance equation; The interfacial stresses, 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 expressions of radial stress and tangential stress to obtain the radial and tangential stresses at any position.

2. A method for obtaining the failure probability of TRISO fuel particles according to claim 1, characterized in that: The method of obtaining the temperature at any position in the TRISO fuel particle model comprises the following steps: The TRISO fuel particle model is divided into n nodes; Obtaining the conduction equation of the TRISO fuel particle model in a spherical coordinate system with an internal heat source, and deriving the temperature equation based on the conduction equation; Obtain the temperature at any radius of the TRISO fuel particle model based on the quasi-steady-state assumption; 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 sequence through the multi-layer structure heat conduction equation.

3. A method for obtaining the failure probability of TRISO fuel particles according to 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 failure probability of TRISO fuel particles according to claim 1, characterized in that: The equations for radial strain and tangential strain are simplified as follows: ； ； in, ； Where, 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 Due to external pressure, F ( t ) is the physical property function required to calculate the creep and swelling deformation caused by irradiation, n is the time increment, a 0 is the coefficient related to swelling strain rate, thermal expansion and creep.

5. A method for obtaining the failure probability of TRISO fuel particles according to claim 4, characterized in that: The failure probability of the TRISO fuel particle model is obtained based on the tensile stress, as shown below: ； Where, f ( t , T )for t time T The failure probability of a node at a given temperature, σ t Due to the combined forces of internal and external forces, T is the node temperature, σ 0 SiC layer strength after irradiation, m is the shape parameter of the Weibull distribution of the SiC layer strength.

6. A TRISO fuel particle failure probability acquisition system, characterized by: include: A construction module is used to construct a TRISO fuel particle model and obtain the temperature, internal pressure generated by the amount of fission gas released, and external pressure at any position in the TRISO fuel particle model; An acquisition module is used to acquire multiple 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 multiple strain types; radial and tangential stresses at any position in the TRISO-type fuel particle model are acquired based on 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; A 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 based on the tensile stress; The strain model is constructed based on multiple strain types, as shown below: ； ； Where, 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 Poisson's ratio, c is the creep coefficient, υ is the Poisson's ratio in creep, S r is the irradiation-induced radial size change rate, is the radial thermal expansion coefficient, t For time, S t is the irradiation-induced tangential size change rate, is the tangential thermal expansion coefficient, is the temperature change rate; The step of obtaining radial and tangential stresses at any position in the TRISO fuel particle model based on the strain model, the temperature at any position in the TRISO fuel particle model, the internal pressure generated by the amount of fission gas released, and the external pressure comprises the following steps: Convert 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 stresses at each interface according to the radial stress balance equation; The interfacial stresses, 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 expressions of radial stress and tangential stress to obtain the radial and tangential stresses at any position.

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