A Reliability Analysis Method for Gas Turbine Blades

Abstract

A gas turbine blade reliability analysis method uses the distribution of uncertainty variables in the gas turbine blade as input to generate a chaotic polynomial to be solved, obtains the coordinates of sample points to be calculated, obtains the corresponding gas turbine blade wall temperature, calculates the gas turbine blade life corresponding to each sample point to be calculated, obtains the expansion coefficients of the chaotic polynomial, and obtains a response surface equation for the gas turbine blade life; generates the coordinates of sampling points required for fitting a Weibull distribution based on the distribution of uncertainty variables, and calculates the gas turbine blade life corresponding to each sampling point; obtains the parameters required for constructing the Weibull distribution, and obtains the probability density function of the Weibull distribution of the gas turbine blade life; and finally calculates the cumulative distribution function, reliability function, risk function, and cumulative risk function of the gas turbine blade life to achieve reliability analysis of the gas turbine blade.

Description

Technical Field

[0001] The invention belongs to the technical field of reliability analysis, and in particular relates to a method for analyzing the reliability of a gas turbine blade. Background Art

[0002] The operating environment of modern gas turbine blades far exceeds the melting point of the turbine metal material. Although protective measures such as film cooling and thermal barrier coatings have been applied to turbine blades, these protection measures are far from sufficient to meet the cooling requirements of the increasingly harsh operating environment. The high failure rate of turbine blades in gas turbines currently in service has become an urgent problem to be addressed. Currently, mainstream gas turbine blade reliability analysis methods rely on deterministic analysis methods, estimating safety factors through extensive experiments and numerical calculations. This presents three problems: 1. The cost of conducting extensive experiments and numerical calculations is prohibitive. 2. Traditional methods cannot utilize existing data as prior knowledge to assist in calculations, resulting in a significant waste of valid information. 3. Reliability issues are inherently uncertain, and relying on deterministic research methods cannot fundamentally understand the mechanisms and guide turbine design. However, the lack of tools for uncertainty-based reliability analysis of gas turbine blades has seriously hindered the development of advanced gas turbines. Summary of the Invention

[0003] To overcome the shortcomings of the prior art, the present invention provides a gas turbine blade reliability analysis method based on a polynomial chaos method and an inverse distance interpolation method. This method helps turbine designers better understand gas turbine blade reliability issues and obtain detailed reliability indicators, thereby guiding the development of advanced gas turbines.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is:

[0005] A gas turbine blade reliability analysis method includes:

[0006] S1, using the distribution of uncertainty variables in the gas turbine blade as input, generating a chaotic polynomial to be solved based on polynomial chaos theory, and calculating the coordinates of sample points to be calculated; the uncertain variables are geometric parameters or aerodynamic parameters that cannot be guaranteed to be equal to the design values during gas turbine operation, and the sample points to be calculated are different gas turbine blade operating conditions that require numerical calculations to solve the chaotic polynomial;

[0007] S2, obtaining the corresponding gas turbine blade wall temperature according to the coordinates of the sample point to be calculated;

[0008] S3, calculating the gas turbine blade life corresponding to each sample point to be calculated according to the gas turbine blade wall temperature;

[0009] S4, calculating the coefficients of the expansion of the chaotic polynomial according to the gas turbine blade life corresponding to each sample point to be calculated, that is, obtaining an explicit expression of the expansion of the chaotic polynomial, the explicit expression being the response surface equation of the gas turbine blade life;

[0010] S5, generating the coordinates of the sampling points required to fit the Weibull distribution according to the distribution of the uncertainty variable;

[0011] S6, calculating the gas turbine blade life corresponding to each sampling point using the response surface equation according to the coordinates of the sampling points;

[0012] S7, calculating parameters required for constructing a Weibull distribution based on the coordinates of the sampling points and the gas turbine blade life corresponding to each sampling point, and obtaining a probability density function of the Weibull distribution of the gas turbine blade life;

[0013] S8, based on the probability density function of the Weibull distribution, calculate the cumulative distribution function (CDF), reliability function (SF), risk function (HF) and cumulative risk function (CHF) of the gas turbine blade life to achieve reliability analysis of the gas turbine blade.

[0014] In one embodiment, the uncertain variables in the gas turbine blade S1 are the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A, which satisfy the normal distribution.

[0015] In one embodiment, S1, the chaotic polynomial to be solved, is expressed as:

[0016]

[0017] Where a0, Represent the orthogonal basis I0, The corresponding coefficient, that is, the quantity that needs to be solved, is the projection of each order, θ is a random variable; in actual operation, the expression is truncated into:

[0018]

[0019] Where y is the system output, i.e. the wall temperature of the gas turbine blade, P is the order of the chaotic polynomial, and a j is the coefficient of the jth orthogonal basis, that is, the coefficient of the orthogonal basis of each order of the chaotic polynomial The discrete form of is the quantity that needs to be solved, Ψ j (ξ) is the jth orthogonal basis in the discrete case;

[0020] The coordinates of the sample points to be calculated are calculated using the Symolyak sparse grid method, and the formula is as follows:

[0021]

[0022] In the formula, n represents the dimension of the problem, that is, the number of types of uncertainty variables, k represents the calculation accuracy, Represents the coordinates of the numerical integration nodes of the n-dimensional k-order sparse grid, q is a constant, q = k + n, |i| = i1 + i2 + i3 + ... + i j +…+i n ,i j Indicates the ordinal number of the one-dimensional numerical integration node of the j-th expansion, j = 1, 2, ..., n, Indicates the ordinal number is i j The node of one-dimensional numerical integration is a node of one-dimensional numerical integration; the numerical integration node is the sample point to be calculated, and each numerical integration node represents a working condition that needs to be calculated to solve the chaotic polynomial.

[0023] In one embodiment, the weight w corresponding to the numerical integration node is calculated by the following formula:

[0024]

[0025] In the formula Indicates the ordinal number is i j The component of the numerical integration node weight, Represents a vector consisting of its components.

[0026] In one embodiment, S2 uses the open source computational fluid dynamics library OpenFOAM to obtain the gas turbine blade wall temperature of the sample point to be calculated, and the method is as follows: the coordinates of each numerical integration node contain n parameters, that is, the uncertainty variables of the working condition represented by the numerical integration node; the n parameters are input into the open source computational fluid dynamics library OpenFOAM to calculate the gas turbine blade wall temperature of the working condition represented by the numerical integration node; the parameters contained in the coordinates of all numerical integration nodes are input into the open source computational fluid dynamics library OpenFOAM to obtain the gas turbine blade wall temperature corresponding to each sample point to be calculated.

[0027] In one embodiment, the S3 first uses an inverse distance interpolation algorithm to obtain a wall temperature-life equation of the gas turbine blade material from the wall temperature-life curve of the gas turbine blade material, and then uses the gas turbine blade wall temperature-life equation and the gas turbine blade wall temperature of each sample point to calculate the gas turbine blade life corresponding to each sample point to be calculated.

[0028] In one embodiment, the expansion coefficients of the chaotic polynomial in S4, i.e., the coefficients of its orthogonal bases of various orders, are solved using the Galerkin projection method, and the formula is as follows:

[0029]

[0030] Where, is the inner product of the polynomial, J(ξ) is the joint probability density function of the uncertainty variables, the coefficients of the chaotic polynomial expansion are matched with the orthogonal bases of various orders I0, That is the explicit expression of the chaotic polynomial expansion required. The input of this explicit expression is the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A of a certain operating condition, and the output is the gas turbine blade life under this operating condition.

[0031] In one embodiment, each coordinate of S5 includes four parameters, namely, the tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0, and the inlet airflow angle A of the sampling point. The number of sampling points Sum is set, and Monte Carlo sampling is used to obtain the coordinates of the sampling points through the following steps:

[0032] 1) First, use Python's open source library random function to randomly generate Sum values distributed between (0,1) and put them into array Z in sequence. Sum middle.

[0033] 2) Let the coordinates of the vth sampling point be (v1, v2, v3, v4), then according to the Monte Carlo principle:

[0034]

[0035]

[0036]

[0037]

[0038] Where Z Sum [v] is the array Z Sum The vth value of γ is the variable for auxiliary calculation. According to the above formula, the coordinates of the vth sampling point can be calculated, that is, the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A of the vth sampling point;

[0039] 3) Perform the operation of step 2) on Sum sampling points to obtain the coordinates of all sampling points.

[0040] In one embodiment, the S7, Weibull distribution satisfies the following formula:

[0041]

[0042] Where τ is the variable of the Weibull distribution probability density function, k w is the Weibull shape parameter, is the quantity to be solved, λ w is the scale parameter of the Weibull distribution and is also the quantity to be solved;

[0043] Calculate k using genetic algorithm w and λ w , the steps are as follows:

[0044] 1) Initialization: Use Python's Random function to randomly generate a pair of k w and λ w , recorded as an individual, repeated p times to obtain an initial population containing p individuals, and the initial evolutionary generation is set to 0;

[0045] 2) Fitness evaluation: According to the k of each individual w and λ w Use the following formula to calculate the mean E of the Weibull distribution represented by this individual w ;

[0046]

[0047] Where η is a variable for auxiliary calculation;

[0048] The mean E of the Weibull distribution represented by the individual w Subtract the mean m of the gas turbine blade life of the Sum sampling points obtained by the sampling point life solution module GE , we can get the fitness of the individual, m GE Calculated by the following formula:

[0049]

[0050] In the formula, Sum is the number of sampling points, life v is the gas turbine blade life at the vth sampling point, which can be obtained in the sampling point life solution module;

[0051] 3) Select operation;

[0052] 4) Crossover operation;

[0053] 5) Mutation operation;

[0054] 6) Randomly generate q individuals to add to the population after mutation operation, q < p;

[0055] 7) Repeat steps 2) to 6) and increase the evolutionary generation by one;

[0056] 8) When the difference in fitness between the best individuals of two adjacent generations is less than 0.00001, the calculation stops. At this time, the individual with the highest fitness in the latest generation is the optimal k w and λ w , the probability density function of the Weibull distribution of gas turbine blade life is obtained.

[0057] In one embodiment, the calculation method of the cumulative distribution function (CDF), reliability function (SF), risk function (HF) and cumulative risk function (CHF) of the gas turbine blade life in S8 is as follows:

[0058]

[0059]

[0060]

[0061]

[0062] Compared with the prior art, the present invention has the following beneficial effects:

[0063] (1) The introduction of the polynomial chaos method can significantly reduce the number of samples required for calculating the life of gas turbine blades. According to the estimation of the embodiment, the number of sample points required to establish the gas turbine blade life response surface using the polynomial chaos method is about 10% of that of the traditional method.

[0064] (2) The inverse distance interpolation module can be used to obtain the temperature-life equation of gas turbine blade materials using the temperature-life relationship of gas turbine blade materials available in the literature, greatly improving the utilization of prior knowledge.

[0065] (3) The reliability analysis method of the entire gas turbine blade is based on the polynomial chaos method, which is a reliability analysis method completely based on the uncertainty theory framework.

[0066] (4) This method can not only calculate the probability density function and cumulative distribution function commonly used in the field of gas turbine blade reliability analysis, but also introduce the reliability function, risk function and cumulative risk function commonly used in reliability analysis in life sciences and social sciences into the field of gas turbine blade reliability analysis, so that turbine designers can have a more comprehensive understanding of the reliability of gas turbine blades. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 Schematic diagram of the blade geometry parameters of the GE-E3 gas turbine.

[0068] Figure 2 Schematic diagram of the system of the present invention.

[0069] Figure 3 This is the wall temperature-life curve of the GE-E3 gas turbine blade material.

[0070] Figure 4 This is the probability density function diagram of the GE-E3 gas turbine blade life.

[0071] Figure 5 This is a reliability function diagram of the GE-E3 gas turbine blade life. DETAILED DESCRIPTION

[0072] The embodiments of the present invention are described in detail below with reference to the accompanying drawings and examples.

[0073] In one embodiment of the present invention, the original thermal field data is from the GE_E3 blade shape (Kwak JS, Han JC. Heat-transfer coefficients of a turbine blade-tip and near-tip regions [J]. Journal of thermophysics and heat transfer, 2003, 17 (3): 297-303.). The geometric parameters of the GE_E3 blade shape are shown in Table 1. The meanings of the various geometric parameters are shown in Table 1. Figure 1 .

[0074] Table 1 Geometric parameters of GE_E3 leaf shape

[0075] Geometry parameter name Value (mm) Tip clearance (S) 0.4 Groove depth (D) 5.08 Shoulder wall thickness (G) 2.29 Blade height (H) 122

[0076] refer to Figure 2 The present embodiment provides a method for analyzing the reliability of a gas turbine blade, which is implemented based on the following functional modules:

[0077] The coordinates of the sample points to be calculated are calculated using the Symolyak sparse grid method.

[0078] 1. Chaotic polynomial model construction and sample point generation module to be calculated

[0079] Taking the distribution of uncertainty variables in gas turbine blades as input, the chaotic polynomial to be solved is generated based on polynomial chaos theory, and the coordinates of the sample points to be calculated are generated.

[0080] In the present invention, uncertainty variables are defined as geometric parameters or aerodynamic parameters that cannot be guaranteed to be equal to the design values during the operation of the gas turbine. Sample points to be calculated are defined as different gas turbine blade operating conditions that require numerical calculations to solve the chaotic polynomial.

[0081] In this embodiment, the tip clearance (S), the mainstream inlet total temperature (T0), the mainstream inlet total pressure (P0) and the inlet airflow angle (A) are selected as uncertainty variables, which respectively satisfy the normal distribution in Table 2.

[0082] This embodiment uses the Symolyak sparse grid method to calculate the coordinates of the sample points to be calculated. The Symolyak sparse grid method uses formula (1) to calculate the numerical integration nodes of the sparse grid accuracy, where the numerical integration nodes are the sample points to be calculated, and each numerical integration node represents a working condition required to solve the chaotic polynomial.

[0083]

[0084] Where n represents the dimension of the problem, that is, the number of types of uncertainty variables, and k represents the calculation accuracy. In this embodiment, n = 4, k = 4; Represents the coordinates of the numerical integration nodes of the n-dimensional k-order sparse grid, q is a constant, q = k + n, |i| = i1 + i2 + i3 + ... + i j +…+i n ,i j Indicates the ordinal number of the one-dimensional numerical integration node of the j-th expansion, j = 1, 2, ..., n, Indicates the ordinal number is i j Node for one-dimensional numerical integration;

[0085] In order to solve the chaotic polynomial, it is also necessary to give the weight w corresponding to each numerical integration node, which can be calculated by formula (2):

[0086]

[0087] In the formula Indicates the ordinal number is i j The component of the numerical integration node weight, Represents a vector composed of its components;

[0088] Therefore, the high-dimensional integral ∫ Ω yΦ j ρ(ξ)dξ can be expressed using the Symolyak sparse grid method as:

[0089]

[0090] Where y is the system output. In this embodiment, y is the wall temperature of the gas turbine blade. j is the integration node of the jth term in the continuous form, ρ(ξ) is the integration weight of the continuous form, N s Indicates the number of nodes for sparse grid numerical integration, y l is the discrete form of y, Φ j (ξl ) is Φ j Discrete form of

[0091] Table 2 Distribution of uncertainty variables

[0092] Geometry parameter name mean Standard deviation Tip clearance (S) 0.4mm 0.08mm Main stream inlet total temperature (D) 709.0K 17.24K Main flow inlet total pressure (G) 126900Pa 7480Pa Inlet airflow angle (H) 0.0° 0.67°

[0093] According to the dimension n and calculation accuracy k of the problem, the chaotic polynomial of the system output (i.e., the wall temperature of the gas turbine blade) y can be established, that is, the chaotic polynomial to be solved, which is expressed as:

[0094]

[0095] Where a0, Represent the orthogonal basis I0, The corresponding coefficient, that is, the quantity that needs to be solved, is the projection of each order, θ is a random variable; in actual operation, the expression is truncated into:

[0096]

[0097] Where P is the order of the chaotic polynomial. The larger the value of P, the more accurate the calculation. In the embodiment, P is set to 4. j is the coefficient of the jth orthogonal basis, that is, the coefficient of the orthogonal basis of each order of the chaotic polynomial The discrete form of Ψ j (ξ) is the jth orthogonal basis in the discrete case.

[0098] 2. Numerical calculation module

[0099] According to the coordinates of the sample point to be calculated, the corresponding gas turbine blade wall temperature is obtained. The gas turbine blade wall temperature of the sample point to be calculated can be obtained using the open source computational fluid dynamics library OpenFOAM. The method is:

[0100] This module accepts the coordinates of the numerical integration nodes of n-dimensional k-order sparse grid accuracy The coordinates of each numerical integration node contain n parameters, which are, in order, the uncertainty variables of the operating condition represented by that numerical integration node. In this example, they are the blade tip clearance (S), the mainstream inlet total temperature (T0), the mainstream inlet total pressure (P0), and the inlet airflow angle (A). Entering these n parameters into the open-source computational fluid dynamics library OpenFOAM calculates the gas turbine blade wall temperature for the operating condition represented by that numerical integration node. Entering the parameters contained in the coordinates of all numerical integration nodes into OpenFOAM yields the gas turbine blade wall temperature corresponding to each sample point to be calculated.

[0101] 3. Inverse distance interpolation module

[0102] The gas turbine blade life corresponding to each sample point to be calculated is calculated based on the gas turbine blade wall temperature. The specific method first uses an inverse distance interpolation algorithm to obtain a wall temperature-life equation for the gas turbine blade material from the wall temperature-life curve of the gas turbine blade material. Then, the gas turbine blade wall temperature-life equation and the gas turbine blade wall temperature at each sample point are used to calculate the gas turbine blade life corresponding to each sample point to be calculated.

[0103] Specifically, this module receives the gas turbine blade wall temperature corresponding to each sample point to be calculated generated by the numerical calculation module. First, according to the GE-E3 gas turbine blade material wall temperature-life curve in the literature (Rowe JP, Freeman JW, Voorhees HR. Final report to the General Electric Company Aircraft Gas Turbine Division on effect of overheating on the creep-rupture properties of Udimet 500alloy at 16000F and 28,500PSI[R]. 1957.), the wall temperature-life equation of the GE-E3 gas turbine blade material is calculated using the inverse distance interpolation method. The wall temperature-life curve of the GE-E3 gas turbine blade material is as follows: Figure 3 shown. Figure 3Point1, Point2, and Point3 are all points with known coordinates given in the literature (Rowe JP, Freeman JW, Voorhees HR. Final report to the General Electric Company Aircraft GasTurbine Division on the effect of overheating on the creep-rupture properties of Udimet 500alloy at 16000F and 28,500PSI[R]. 1957.). The coordinates of Point1, Point2, and Point3 are (0.0035, 2000), (0.1210, 1900), and (16.374, 1800), respectively. The first law of geography applies: all attribute values on a geographic surface are interrelated, but closer values are more correlated than farther values. Combining the coordinates of Point1, Point2, and Point3, the gas turbine blade life corresponding to any gas turbine blade wall temperature can be calculated:

[0104]

[0105] In the formula, life any is the gas turbine blade life at any gas turbine blade wall temperature, d tem1 ,d tem2 ,d tem3 The difference between the wall temperature of the gas turbine blade and the horizontal coordinates of Point1, Point2, and Point3. life1, life2, and life3 are the lifespans of Point1, Point2, and Point3, respectively. The numerical calculation module receives the wall temperature of the gas turbine blade corresponding to each sample point to be calculated and uses formula (6) to calculate the gas turbine blade life corresponding to each sample point to be calculated.

[0106] 4. Chaotic polynomial model solving module

[0107] According to the gas turbine blade life corresponding to each sample point to be calculated, the expansion coefficient of the chaotic polynomial is obtained, that is, the explicit expression of the expansion of the chaotic polynomial is obtained. The explicit expression is the response surface equation of the gas turbine blade life.

[0108] Specifically, this module receives the gas turbine blade life corresponding to each sample point to be calculated generated by the inverse distance interpolation module, so that the Galerkin projection method can be used to obtain the chaotic polynomial model building and the chaotic polynomial to be solved in the sample point generation module. The corresponding coefficients are used to establish the explicit expression of the chaotic polynomial equation, which is the response surface equation for the life of the GE-E3 gas turbine blade. The coefficients of the chaotic polynomial expansion are the coefficients of its orthogonal basis of each order, which are solved using the Galerkin projection method. The formula is as follows:

[0109]

[0110] Where, j (ξ) represents the jth orthogonal basis, is the inner product of the polynomial, J(ξ) is the joint probability density function of the uncertain input variables, the coefficients of the chaotic polynomial expansion are matched with the orthogonal bases I0, I1 and I2 of the polynomials in formula (4). This is the explicit expression for the desired chaotic polynomial expansion. The inputs to this explicit expression are the blade tip clearance (S), mainstream inlet total temperature (T0), mainstream inlet total pressure (P0), and inlet airflow angle (A) for a specific operating condition. The output is the GE-E3 gas turbine blade life under that operating condition. This explicit expression also serves as the response surface equation for the GE-E3 gas turbine blade life.

[0111] 5. Monte Carlo Sampling Module

[0112] According to the distribution of uncertainty variables, the coordinates of the sampling points required to fit the Weibull distribution are generated.

[0113] Specifically, each coordinate contains four parameters: the tip clearance (S), the mainstream inlet total temperature (T0), the mainstream inlet total pressure (P0), and the inlet airflow angle (A) at that sampling point. To fit the Weibull distribution, the number of sampling points (Sum) needs to be set. The larger the value, the better. In this embodiment, the number of sampling points (Sum) is set to 100,000. The steps for obtaining the coordinates of these 100,000 sampling points using Monte Carlo sampling are as follows:

[0114] 1) First, use Python's open source library random function to randomly generate Sum values distributed between (0,1) and put them into array Z in sequence. Sum middle.

[0115] 2) Let the coordinates of the vth sampling point be (v1, v2, v3, v4), then according to the Monte Carlo principle:

[0116]

[0117]

[0118]

[0119]

[0120] Where Z Sum [v] is the array Z Sum The vth value of γ. γ is a variable used in the auxiliary calculation. According to formulas (8, 9, 10, 11), the coordinates of the vth sampling point can be calculated, namely the blade tip clearance (S), mainstream inlet total temperature (T0), mainstream inlet total pressure (P0), and inlet airflow angle (A) at the vth sampling point.

[0121] 3) Perform step (2) on all Sum sampling points to obtain the coordinates of all these sampling points.

[0122] 6. Sampling point life calculation module

[0123] According to the coordinates of the sampling points, the response surface equation is used to calculate the gas turbine blade life corresponding to each sampling point.

[0124] Specifically, this module receives the coordinates of 100,000 sampling points generated by the Monte Carlo sampling module and inputs them into the response surface equation of the GE-E3 gas turbine blade life obtained in the chaotic polynomial model solving module to calculate the GE-E3 gas turbine blade life corresponding to each sampling point.

[0125] 7. Weibull distribution fitting module

[0126] According to the coordinates of the sampling points and the gas turbine blade life corresponding to each sampling point, the parameters required to construct the Weibull distribution are obtained, and the probability density function of the Weibull distribution of the gas turbine blade life is obtained.

[0127] Specifically, this module receives the sampling point coordinates generated by the Monte Carlo sampling module and the GE-E3 gas turbine blade life corresponding to each sampling point generated by the sampling point life solution module. It uses the genetic algorithm to obtain the parameters required to construct the Weibull distribution and obtain the probability density function of the Weibull distribution of the GE-E3 gas turbine blade life. The Weibull distribution satisfies formula (12):

[0128]

[0129] Where τ is the variable of the Weibull distribution probability density function, k w is the Weibull shape parameter, is the quantity to be solved, λ w is the scale parameter of the Weibull distribution and is also the quantity to be solved. It can be found that solving k w and λ w The probability density function of the Weibull distribution of the life of the gas turbine blade can be converted into an explicit calculation formula. The present invention innovatively introduces a genetic algorithm to solve k w and λ wThe speed and accuracy of solving the Weibull distribution are greatly improved by the present invention. w and λ w The steps are as follows:

[0130] 1) Initialization: Use Python's Random function to randomly generate a pair of k w and λ w , recorded as an individual, repeated 100 times to obtain an initial population containing 100 individuals, and the initial evolutionary generation is set to 0;

[0131] 2) Fitness evaluation: According to the k of each individual w and λ w Use formula (13) to calculate the mean E of the Weibull distribution represented by the individual w ;

[0132]

[0133] Where η is a variable used for auxiliary calculation.

[0134] The mean E of the Weibull distribution represented by the individual w Subtract the mean value m of the GE-E3 gas turbine blade life of 100,000 sampling points obtained by the sampling point life calculation module GE , we can get the fitness of the individual, m GE It can be calculated by formula (14):

[0135]

[0136] Where Sum is the number of sampling points, which is set to 100,000 in this embodiment. v is the GE-E3 gas turbine blade life at the vth sampling point, which can be obtained in the sampling point life solution module.

[0137] 3) Selection operation: select the top 60% of individuals in fitness to enter the crossover operation, and eliminate the bottom 40% of individuals;

[0138] 4) Crossover operation: Randomly swap the codes of the individuals obtained by the selection operation with a probability of 85%;

[0139] 5) Mutation operation: randomly replace the code of the individual obtained by the crossover operation with a random number with a probability of 2%;

[0140] 6) Randomly generate 40 individuals to supplement the population after the mutation operation;

[0141] 7) Repeat steps 2) to 6) and increase the evolutionary generation by one;

[0142] 8) When the difference in fitness between the best individuals of two adjacent generations is less than 0.00001, the calculation stops. At this time, the individual with the highest fitness in the latest generation is the optimal k w and λ w At this point, the probability density function of the Weibull distribution of the GE-E3 gas turbine blade life can be obtained.

[0143] 8. Reliability result calculation module

[0144] According to the probability density function of Weibull distribution, the cumulative distribution function (CDF), reliability function (SF), risk function (HF) and cumulative risk function (CHF) of gas turbine blade life are calculated to realize the reliability analysis of gas turbine blades.

[0145] The calculation methods of CDF, SF, HF, and CHF are as follows;

[0146]

[0147]

[0148]

[0149]

[0150] Figure 4 The probability density function plot for the blade life of a GE-E3 gas turbine is shown in Figure 1. As can be seen from the figure, under the influence of uncertain variables, the mean life of the GE-E3 gas turbine decreases rapidly, reaching approximately 20% of the design operating condition. This indicates that in the manufacturing process of advanced gas turbines, the geometric accuracy of the blade tip clearance must be strictly guaranteed, and a suitable active control system must be designed to rapidly attenuate fluctuations in the mainstream inlet total temperature, mainstream inlet total pressure, and inlet flow angle during actual turbine operation.

[0151] Figure 5 This is a reliability function graph for the life of a GE-E3 gas turbine blade. The vertical axis represents the survival number, or the number of blades in the GE-E3 gas turbine that are operating well. It can be seen that as the life (operating time) increases, the number of blades in the GE-E3 gas turbine that are operating well decreases rapidly. Figure 5This indicates that a gas turbine should undergo a comprehensive overhaul approximately 20% of the time when it is operating under design conditions, as only 60% of the blades are still functioning properly, severely impacting the turbine's performance. It is noteworthy that neither the probability density function nor the reliability function of the GE-E3 gas turbine blade lifespan are currently available in domestic or international literature, and are virtually impossible to obtain using existing deterministic reliability analysis methods. The gas turbine blade reliability analysis method presented in this invention significantly broadens turbine designers' understanding of turbine robustness and can provide quantitative conclusions on when a gas turbine should undergo an overhaul during actual operation. Therefore, this invention is of great significance to the manufacture of advanced gas turbines.

Claims

1. A method for analyzing the reliability of a gas turbine blade, characterized in that: include: S1, using the distribution of uncertainty variables in the gas turbine blade as input, generating a chaotic polynomial to be solved based on polynomial chaos theory, and calculating the coordinates of sample points to be calculated; the uncertain variables are geometric parameters or aerodynamic parameters that cannot be guaranteed to be equal to the design values during gas turbine operation, and the sample points to be calculated are different gas turbine blade operating conditions that require numerical calculations to solve the chaotic polynomial; S2, obtaining the corresponding gas turbine blade wall temperature according to the coordinates of the sample point to be calculated; S3, calculating the gas turbine blade life corresponding to each sample point to be calculated according to the gas turbine blade wall temperature; S4, according to the gas turbine blade life corresponding to each sample point to be calculated, calculating the expansion coefficient of the chaotic polynomial, that is, obtaining an explicit expression of the expansion of the chaotic polynomial, the explicit expression being the response surface equation of the gas turbine blade life; S5, generating the coordinates of the sampling points required to fit the Weibull distribution according to the distribution of the uncertainty variable; S6, calculating the gas turbine blade life corresponding to each sampling point using the response surface equation according to the coordinates of the sampling points; S7, calculating parameters required for constructing a Weibull distribution based on the coordinates of the sampling points and the gas turbine blade life corresponding to each sampling point, and obtaining a probability density function of the Weibull distribution of the gas turbine blade life; S8, based on the probability density function of the Weibull distribution, calculate the cumulative distribution function (CDF), reliability function (SF), risk function (HF) and cumulative risk function (CHF) of the gas turbine blade life to achieve reliability analysis of the gas turbine blade.

2. The gas turbine blade reliability analysis method according to claim 1, characterized in that: The uncertain variables in S1, the gas turbine blades, are the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A, which satisfy the normal distribution.

3. The gas turbine blade reliability analysis method according to claim 1 or 2, characterized in that: Said S1, the chaotic polynomial to be solved, is expressed as: Where a0, Represent the orthogonal basis I0, The corresponding coefficient, that is, the quantity that needs to be solved, is the projection of each order, θ is a random variable; in actual operation, the expression is truncated into: Where y is the system output, i.e., the wall temperature of the gas turbine blade, P is the order of the chaotic polynomial, and a j is the coefficient of the jth orthogonal basis, that is, the coefficient of the orthogonal basis of each order of the chaotic polynomial The discrete form of is the quantity that needs to be solved, Ψ j (ξ) is the jth orthogonal basis in the discrete case; The coordinates of the sample points to be calculated are calculated using the Symolyak sparse grid method, and the formula is as follows: In the formula, n represents the dimension of the problem, that is, the number of types of uncertainty variables, k represents the calculation accuracy, Represents the coordinates of the numerical integration nodes of the n-dimensional k-order sparse grid, q is a constant, q = k + n, |i| = i1 + i2 + i3 + ... + i j +…+i n ,i j Indicates the ordinal number of the one-dimensional numerical integration node of the j-th expansion, j = 1, 2, ..., n, Indicates the ordinal number is i j The node of one-dimensional numerical integration is a node of one-dimensional numerical integration; the numerical integration node is the sample point to be calculated, and each numerical integration node represents a working condition that needs to be calculated to solve the chaotic polynomial.

4. The gas turbine blade reliability analysis method according to claim 3, characterized in that: The weight w corresponding to the numerical integration node is calculated by the following formula: In the formula Indicates the ordinal number is i j The component of the numerical integration node weight, Represents a vector consisting of its components.

5. The gas turbine blade reliability analysis method according to claim 3, characterized in that: The S2 uses the open source computational fluid dynamics library OpenFOAM to obtain the gas turbine blade wall temperature of the sample point to be calculated, and the method is as follows: the coordinates of each numerical integration node contain n parameters, that is, the uncertainty variables of the working condition represented by the numerical integration node; the n parameters are input into the open source computational fluid dynamics library OpenFOAM to calculate the gas turbine blade wall temperature of the working condition represented by the numerical integration node; the parameters contained in the coordinates of all numerical integration nodes are input into the open source computational fluid dynamics library OpenFOAM to obtain the gas turbine blade wall temperature corresponding to each sample point to be calculated.

6. The gas turbine blade reliability analysis method according to claim 5, characterized in that: The S3 first uses an inverse distance interpolation algorithm to obtain a wall temperature-life equation of the gas turbine blade material from the wall temperature-life curve of the gas turbine blade material, and then uses the gas turbine blade wall temperature-life equation and the gas turbine blade wall temperature of each sample point to calculate the gas turbine blade life corresponding to each sample point to be calculated.

7. The gas turbine blade reliability analysis method according to claim 6, characterized in that: The expansion coefficients of the chaotic polynomial S4, i.e., the coefficients of its orthogonal bases of various orders, are solved using the Galerkin projection method, and the formula is as follows: Where, is the inner product of the polynomial, J(ξ) is the joint probability density function of the uncertainty variables, the coefficients of the chaotic polynomial expansion are matched with the orthogonal bases of various orders I0, That is the explicit expression of the chaotic polynomial expansion required. The input of this explicit expression is the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A of a certain operating condition, and the output is the gas turbine blade life under this operating condition.

8. The gas turbine blade reliability analysis method according to claim 7, characterized in that: Each coordinate of S5 contains four parameters, namely the tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0, and the inlet airflow angle A of the sampling point. The number of sampling points Sum is set, and Monte Carlo sampling is used to obtain the coordinates of the sampling points through the following steps: 1) First, use Python's open source library random function to randomly generate Sum values distributed between (0,1) and put them into array Z in sequence. Sum middle; 2) Let the coordinates of the vth sampling point be (v1, v2, v3, v4), then according to the Monte Carlo principle: Where Z Sum [v] is the array Z Sum The vth value of γ is the variable for auxiliary calculation. According to the above formula, the coordinates of the vth sampling point can be calculated, that is, the blade tip clearance S, the mainstream inlet total temperature T0, the mainstream inlet total pressure P0 and the inlet airflow angle A of the vth sampling point; 3) Perform the operation of step 2) on Sum sampling points to obtain the coordinates of all sampling points.

9. The gas turbine blade reliability analysis method according to claim 8, characterized in that: The S7, Weibull distribution satisfies the following formula: Where τ is the variable of the Weibull distribution probability density function, k w is the Weibull shape parameter, is the quantity to be solved, λ w is the scale parameter of the Weibull distribution and is also the quantity to be solved; Calculate k using genetic algorithm w and λ w , the steps are as follows: 1) Initialization: Use Python's Random function to randomly generate a pair of k w and λ w , recorded as an individual, repeated p times to obtain an initial population containing p individuals, and the initial evolutionary generation is set to 0; 2) Fitness evaluation: According to the k of each individual w and λ w Use the following formula to calculate the mean E of the Weibull distribution represented by this individual w ; Where η is a variable for auxiliary calculation; The mean E of the Weibull distribution represented by the individual w Subtract the mean m of the gas turbine blade life of the Sum sampling points obtained by the sampling point life solution module GE , we can get the fitness of the individual, m GE Calculated by the following formula: In the formula, Sum is the number of sampling points, life v is the gas turbine blade life at the vth sampling point, which can be obtained in the sampling point life solution module; 3) Select operation; 4) Crossover operation; 5) Mutation operation; 6) Randomly generate q individuals to add to the population after mutation operation, q < p; 7) Repeat steps 2) to 6) and increase the evolutionary generation by one; 8) When the difference in fitness between the best individuals of two adjacent generations is less than 0.00001, the calculation stops. At this time, the individual with the highest fitness in the latest generation is the optimal k w and λ w , the probability density function of the Weibull distribution of gas turbine blade life is obtained.

10. The gas turbine blade reliability analysis method according to claim 9, characterized in that: The calculation methods of the cumulative distribution function (CDF), reliability function (SF), risk function (HF) and cumulative risk function (CHF) of the gas turbine blade life in S8 are as follows:

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