A reduced-order analysis method for thermal stratification of the lead pool of a lead-bismuth fast reactor

Abstract

The present invention proposes a reduced-order analysis method for thermal stratification in the lead pool of a lead-bismuth fast reactor, including obtaining a full-order snapshot of the temperature field based on CFD numerical simulation technology; extracting characteristic orthogonal bases based on the full-order snapshot and determining the number of basis functions based on the energy contribution rate; obtaining a reduced-order model using the Galerkin reduced-order projection technique; reconstructing the axial and radial temperature fields of the lead pool using the reduced-order model, which is in good agreement with the full-order snapshot temperature field; and finally, conducting a sensitivity analysis on the number of selected basis functions. The accuracy of the reduced-order model is further improved with increasing the number of basis functions, verifying its numerical convergence. The reduced-order model proposed in the present invention can well simulate the temperature distribution in the upper chamber and can quickly study the thermal stratification interface characteristics under lead-bismuth fast reactor accidents. This provides a good reference for studying the mechanism of thermal stratification and effectively curbing its occurrence.

Description

Technical Field

[0001] The invention belongs to the field of complex thermal hydraulic phenomena in reactors, and in particular relates to an analysis method for a lead pool thermal stratification model of a lead-bismuth fast reactor. Background Art

[0002] Thermal stratification (TS) plays a vital role in the safety of nuclear reactor systems. After a lead-bismuth fast reactor is shut down due to an accident, the lead-bismuth flow rate at the core outlet decreases rapidly according to an approximately exponential relationship, and the lead-bismuth temperature at the core outlet drops rapidly. In the outlet chamber, the low-temperature, low-speed fluid does not have enough inertia to rush into a higher position and fully mix with the high-temperature lead and bismuth at a higher position, so it can only enter the bottom of the chamber, resulting in thermal stratification. The occurrence of thermal stratification in the outlet chamber will cause significant thermal stress in the internal structure of the chamber, causing thermal fatigue mechanical damage. It will also affect the establishment of natural circulation of the reactor after shutdown and affect the discharge of residual heat. [1] Therefore, a detailed understanding of the thermal stratification phenomenon is crucial for the optimal design and safe operation of fast reactors.

[0003] For decades, efforts have focused on simulating thermal stratification in reactors to prevent or mitigate the damage caused by this phenomenon. A thorough review of recent advances in computational simulation methods for thermal stratification in reactors reveals that these approaches can generally be categorized into three categories. The first is experimental methods, which involve constructing test benches or experimental reactors to experimentally analyze thermal stratification and other thermal-hydraulic phenomena. The second is system-based approaches, which offer fast but approximate calculations. The third is computational fluid dynamics (CFD) methods, which offer high-resolution and high-accuracy calculations but are computationally expensive and time-consuming. Therefore, when extensive transient calculations are required, fast-running reduced-order thermal stratification models with higher fidelity are essential for reactor development, core safety analysis, and reactor licensing. After validation, these models are implemented into system-level codes, facilitating fast computations of the entire system. When coupled with CFD codes for more accurate calculations, they can reduce the number of iterations between the two codes and shorten computational time. Summary of the Invention

[0004] To address the limitations of existing traditional lead pool thermal stratification analysis methods, this paper proposes a reduced-order analysis method for lead pool thermal stratification in a lead-bismuth fast reactor. This method involves establishing an appropriate mathematical and physical model, generating high-precision full-order snapshots of the temperature field using Fluent numerical simulation, selecting time segments of the temperature field using these snapshots, reducing the order of the thermal stratification model based on the full-order snapshots, and reconstructing the temperature field data based on the reduced-order model. The developed reduced-order model demonstrates its superior computational efficiency and accuracy.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A thermal stratification reduction analysis method for a lead pool of a lead-bismuth fast reactor proposed in the present invention comprises the following steps:

[0007] Step 1: Establish a mathematical and physical model of lead pool thermal stratification;

[0008] Step 2: Perform numerical simulation based on computational fluid dynamics (CFD);

[0009] Step 3: Selecting temperature field data based on time snapshots;

[0010] Step 4: Construct a thermal layered reduced-order model based on Galerkin projection for the full-order system;

[0011] Step 5: Reconstruct the temperature field data based on the reduced-order model.

[0012] The detailed steps of step 1 are as follows:

[0013] Step 1: Select the structural parameters and experimental parameters based on the similarity of the Richardson number (Ri) dimensional analysis. The Richardson number characterizes the degree of thermal stratification. Its practical significance lies in the ratio of the buoyancy force to the inertial force. That is, when the buoyancy force plays a significant role relative to the inertial force of the fluid, the lead-bismuth fluid will experience thermal stratification. Ri can be expressed as the ratio of the Grashof number (Gr) to the square of the Reynolds number (Re).

[0014] (12)

[0015] (13)

[0016] (14)

[0017] in, is the dynamic viscosity, Vc is the characteristic velocity, g is the acceleration due to gravity, β is the thermal expansion coefficient, , where ρ c is the low temperature density, ρ p is the high temperature density, △T is the average axial temperature gradient, L is the characteristic length, V is the characteristic velocity, Buoyancy represents the buoyancy force, and Intertia represents the inertial force;

[0018] Step 2: Establish the heat transfer control equation of the lead pool thermal stratification:

[0019] In the model establishment process, the liquid lead-bismuth fluid is regarded as an incompressible single-phase turbulent flow. The mass, momentum, and energy equations within the control system can be expressed as follows:

[0020] (15)

[0021] (16)

[0022] (17)

[0023] Among them, u represents speed, x represents distance, i and j represent different directions, ρ represents density, t represents time, p represents pressure, μ represents viscosity, λ represents thermal conductivity, Cp represents specific heat capacity, and T represents temperature. represents the turbulent additional stress expressed in average form, represents the turbulent additional heat flux in average form;

[0024] For the complex chamber turbulent flow in the present invention, the k-ε model, which is widely used in engineering, is adopted. The eddy viscosity model and the Boussinesq approximation hypothesis are introduced, and the eddy viscosity viscosity is introduced. and turbulent thermal conductivity , then the momentum and energy equations for this experimental condition can be expressed as:

[0025] (18)

[0026] (19)

[0027] Where v represents the dynamic viscosity, λ represents the thermal conductivity, and v t represents the turbulent additional viscosity, λ t represents the turbulent additional thermal conductivity.

[0028] The detailed steps of step 2 of the reduced-order analysis are as follows:

[0029] Step 1: Use Fluent software to build a three-dimensional geometric model of the lead pool, fully consider the influence of intermediate structural parts, and determine the inlet and outlet boundary conditions.

[0030] Step 2: Grid the model, perform grid non-uniformization on the boundary layer grid and the complex streamline area, and perform grid independence analysis and time gradient independence verification.

[0031] Step 3: Establish the boundary conditions for the lead pool inlet and outlet, using a turbulence model suitable for lead-bismuth fluids. Select an appropriate numerical processing format for the discretization of time and space terms.

[0032] Step 4: Calculate the accurate temperature field distribution of the lead pool fluid domain and obtain a full-order snapshot.

[0033] In step 3 of the order reduction analysis, based on the full-order data obtained by CFD, a time series snapshot is selected at a specific time point; a time-space matrix is formed using the time snapshot and the temperature field data at the corresponding time;

[0034] The detailed steps of step 4 of the reduced-order analysis are as follows:

[0035] Step 1: Get the full-order snapshot of the temperature field through numerical simulation, and take the temperature field calculated by numerical calculation for L time periods, which is recorded as ;

[0036] (20)

[0037] Among them, M represents a discrete point in space, and N represents a discrete point in time;

[0038] Step 2: Remove the average signal component before performing proper orthogonal decomposition (POD). This needs to be reconsidered when reconstructing the field data. The average value of a discrete point in space based on the temperature snapshot matrix can be expressed as:

[0039] (twenty one)

[0040] Step 3: Use the singular value decomposition (SVD) method to find the standard eigenvector v and the corresponding eigenvalue λ of the matrix. At this time, the eigenvalues are arranged from large to small.

[0041] (twenty two)

[0042] Among them, V is the modal matrix, each column of which corresponds to a basis function; U is an M×M unitary matrix, S is an M×N non-negative real symmetric matrix, the elements of its diagonal are singular values, and the time coefficient matrix , each column of which corresponds to the time constant of the mode; the modal energy can be obtained by calculating the singular value, that is, ;

[0043] Step 4: Select the number of basis functions according to the energy contribution rate, take the first M eigenvalues and corresponding eigenvectors, and generate POD basis functions based on the eigenvalues and eigenvectors;

[0044] (twenty three)

[0045] Step 5: Determine whether the approximate solution of the temperature field meets the accuracy requirements. If not, return to the first step to reconstruct the full-order snapshot and then select the POD basis function until the error accuracy requirements are met.

[0046] The detailed steps of step 5 of the reduced-order analysis are as follows:

[0047] Step 1: Based on the Galerkin method, the full-order temperature field data is subjected to low-dimensional linear combination by selecting the number of basis functions according to the energy contribution rate:

[0048] (twenty four)

[0049] in, is the mode (i.e. basis function), a i (t) is the time constant of the corresponding mode;

[0050] Step 2: Add the average value of step 2 in step 4 to reconstruct the temperature field data.

[0051] (25)

[0052] Step 3: Perform error analysis on the temperature field reconstructed by the data and the full-order temperature field obtained by the CFD method to minimize the error between the POD solution of the temperature field and the solution of the traditional numerical method:

[0053] (26)

[0054] in, is the numerical solution of the temperature field obtained by the numerical method, T(x,y,z) is the reduced-order solution of the temperature field obtained by the POD method, Represents the square of the absolute value.

[0055] The present invention proposes a method for reducing the order of thermal stratification in the lead pool of a lead-bismuth fast reactor. It uses Fluent numerical simulation to obtain a high-precision full-order snapshot of the temperature field. Based on the characteristic orthogonal basis decomposition, a limited number of basis functions are selected to develop a thermal stratification reduction model. The developed reduction model can better predict the temperature distribution of the chamber and has a good predictive effect on the thermal stratification interface characteristics. The selection of the POD basis function of the present invention is based on the full-order snapshot of CFD numerical simulation. It can also be calculated and measured by other numerical methods or experimental methods. Therefore, the present invention also has a certain ability to generalize to other data. The lead pool thermal stratification reduction model of the present invention greatly shortens the calculation time and saves computing resources while ensuring sufficient accuracy, and realizes the rapid prediction of the temperature distribution at the thermal stratification. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1a , Figure 1b The diagram of the lead pool thermal layered structure model and grid division is shown in Figure 2. Figure 1a is the model structure, Figure 1b For mesh division;

[0057] Figure 2 Schematic diagram of basis function energy contribution ratio;

[0058] Figure 3Schematic diagram of error analysis corresponding to basis functions;

[0059] Figure 4a , Figure 4b is a schematic diagram of the reconstruction of the axial temperature field; Figure 4a Reconstruction of the 5s axial temperature distribution, Figure 4b Reconstruction of axial temperature distribution for 10s;

[0060] Figure 5 The present invention provides a flow chart of a thermal stratification reduction analysis method for a lead pool of a lead-bismuth fast reactor. Specific implementation methods

[0061] To make the key implementation contents of the present invention more specific and clear, the following is a complete and clear description of the specific implementation methods and specific implementation objects and methods of the present invention. Based on the implementation examples of the present invention, other implementation examples produced by other persons in the field without engaging in creative activities should also fall within the scope of protection supported by the present invention.

[0062] To describe the present invention in more detail, a detailed description is given below with reference to the accompanying drawings and implementation examples.

[0063] like Figure 5 As shown, a thermal stratification reduction analysis method for a lead pool of a lead-bismuth fast reactor of the present invention specifically comprises the following steps:

[0064] Step 1: Establish a mathematical and physical model of lead pool thermal stratification:

[0065] For this embodiment, the lead pool thermal stratification analysis model is as follows: Figure 1a , Figure 1b The experimental and structural parameters are shown in Table 1, with the thermophysical properties of lead and bismuth shown in Table 2. It should be noted that for the selected lead pool thermal stratification analysis model, the structural and experimental parameters were selected based on the similarity of the Richardson number (Ri) dimensional analysis. The Richardson number characterizes the degree of thermal stratification. Its practical significance lies in the ratio of the buoyancy force to the inertial force. Specifically, when the buoyancy force of a lead-bismuth fluid becomes significant relative to the inertial force, thermal stratification will occur. Ri is expressed as the ratio of the Grashof number to the square of the Reynolds number.

[0066] Table 1 Model structure parameters and experimental parameters

[0067]

[0068] Table 2 Thermophysical property relationship of lead-bismuth fluid

[0069]

[0070] Step 2: Perform numerical simulation based on computational fluid dynamics (CFD), including:

[0071] Step 2.1: Use Fluent software to perform geometric modeling for the mathematical and physical model established in step 1, and use structured grid and non-uniform grid technology for grid division;

[0072] Step 2.2: Perform grid independence test on the divided grid to ensure that the influence of grid differences on the calculation results is minimal, and verify the time gradient independence;

[0073] Step 2.3: Set up the Fluent solver for the given boundary conditions, with the bottom inlet as the inlet mass flow boundary, the outlet as the pressure boundary, and no-slip boundaries at the walls. Based on the thermophysical properties of the lead-bismuth fluid, the model selection and numerical solution method in the Fluent solver are set as follows: the lead-bismuth fluid is an incompressible viscous Newtonian fluid, the standard Ke model is used as the turbulence model, the standard wall function is used for near-wall treatment, the Poisson (Bounssinesq) approximation is adopted, and the pressure-velocity coupling calculation uses the PIMPLE algorithm, which combines the semi-implicit method (SIMPLE) and the pressure-implicit splitting operator method (PISO). This example does not consider thermo-solid coupling. The transient term, convection term, and diffusion term in the governing equations are discretized using the fully implicit algorithm, the second-order upwind difference scheme (QUICK), and the central difference algorithm, respectively.

[0074] Step 3: Temperature field data selection based on time snapshots. Based on the full-order data obtained by CFD, time series snapshots are selected at specific time points, and the time snapshots and the temperature field data at the corresponding time are used to form a time-space matrix.

[0075] Step 4: Construct a thermal layered reduced-order model based on the Galerkin projection for the full-order system, including:

[0076] Step 4.1: Get the full-order snapshot of the temperature field through numerical simulation. Take the temperature field calculated by numerical calculation for L time periods and record it as ;

[0077] (20)

[0078] Among them, M represents a discrete point in space, and N represents a discrete point in time;

[0079] Step 4.2: Remove the average signal component before performing the proper orthogonal basis decomposition (POD) decomposition. This needs to be reconsidered when reconstructing the field data. The average value of a discrete point in space based on the temperature snapshot matrix can be expressed as:

[0080] (twenty one)

[0081] Step 4.3: Use the SVD method to find the standard eigenvector v and the corresponding eigenvalue λ of the matrix. At this time, the eigenvalues are arranged from large to small.

[0082] (twenty two)

[0083] Among them, V is the modal matrix, each column of which corresponds to a basis function; U is an M×M unitary matrix, S is an M×N non-negative real symmetric matrix, the elements of its diagonal are singular values, and the time coefficient matrix , each column of which is the time constant of the mode; the modal energy can be obtained by calculating the singular value, .

[0084] Step 4.4: Select the number of basis functions according to the energy contribution rate, take the first M eigenvalues and corresponding eigenvectors, and generate POD basis functions based on the eigenvalues and eigenvectors;

[0085] (twenty three)

[0086] Among them, M represents the number of eigenvalues taken, and N represents the dimension of the full-order matrix.

[0087] Step 4.5: Determine whether the approximate solution of the temperature field meets the accuracy requirements. If not, return to the first step to reconstruct the full-order snapshot and then select the POD basis function until the error accuracy requirements are met.

[0088] Step 5: Reconstruct the temperature field data based on the reduced-order model, specifically including:

[0089] Step 5.1: Based on the Galerkin method, select the number of basis functions according to the energy contribution rate to perform low-dimensional linear combination of the full-order temperature field data:

[0090] (twenty four)

[0091] in, is the mode (i.e. basis function), a i (t) is the time constant of the corresponding mode;

[0092] Step 5.2: Add the average value in step 4.2 to reconstruct the temperature field data:

[0093] (25)

[0094] Step 5.3: Perform error analysis on the temperature field reconstructed from the data and the full-order temperature field obtained by the CFD method to minimize the error between the POD solution of the temperature field and the solution of the traditional numerical method:

[0095] (26)

[0096] in, is the numerical solution of the temperature field obtained by the numerical method, T(x,y,z) is the reduced-order solution of the temperature field obtained by the POD method, Represents the square of the absolute value.

[0097] This implementation example uses Fluent numerical simulation to obtain a high-precision full-order snapshot of the temperature field. Based on the characteristic orthogonal basis decomposition, a limited number of basis functions (5) are selected to develop a thermal layered reduced-order model. When the basis function is 5, the reduced-order analysis is performed, such as Figure 2 The energy contribution rate shown is close to 100%, as shown in Figure 3 As shown in the figure, the temperature error is within 2K, which meets the accuracy requirement. The developed reduced-order model can better predict the temperature distribution in the chamber and has a good predictive effect on the thermal stratification interface characteristics, such as Figure 4a , Figure 4b shown.

[0098] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for thermal stratification reduction analysis of a lead pool in a lead-bismuth fast reactor, characterized in that: The steps include: Step 1: Establish a mathematical and physical model of lead pool thermal stratification, including: Step 1: Selection of similar deformed structural parameters and experimental parameters based on the Richardson number Ri dimensional analysis: Ri is expressed as the ratio of the Grashof number Gr to the square of the Reynolds number Re: (14) Where g is the acceleration due to gravity, β is the coefficient of thermal expansion, , where ρ c is the low temperature density, ρ p is the high temperature density, △T is the average axial temperature gradient, L is the characteristic length, V is the characteristic velocity; Buoyancy represents the buoyancy force, and Intertia represents the inertial force; Step 2: Establish the heat transfer control equation of the lead pool thermal stratification: The mass, momentum, and energy equations within the control system are expressed as follows: (15) (16) (17) Among them, u represents speed, x represents distance, i and j represent different directions, ρ represents density, t represents time, p represents pressure, μ represents viscosity, λ represents thermal conductivity, Cp represents specific heat capacity, and T represents temperature. represents the turbulent additional stress expressed in average form, represents the turbulent additional heat flux in average form; Step 2: Perform numerical simulation based on computational fluid dynamics, including: Step 1: Create a three-dimensional geometric model of the lead pool, fully consider the influence of the intermediate structural parts, and determine the inlet and outlet boundary conditions; Step 2: Mesh the model and perform mesh independence analysis and verification; Step 3: Establish the boundary conditions at the lead pool inlet and outlet, using the lead-bismuth fluid turbulence model; select the numerical processing format for the discretization of time and space terms; Step 4: Calculate the accurate temperature field distribution of the lead pool fluid domain and obtain a full-order snapshot; Step 3: Temperature field data selection based on time snapshots, including: Step 1: Select time series snapshots based on the full-order data obtained from computational fluid dynamics; Step 2: Use the time snapshot and the temperature field data at the corresponding time to form a time-space matrix; Step 4: Construct a thermal layered reduced-order model based on Galerkin projection for the full-order system, including: Step 1: Get the full-order snapshot of the temperature field through numerical simulation, and take the temperature field calculated by numerical calculation for L time periods, which is recorded as ; (20) Among them, M represents a discrete point in space, and N represents a discrete point in time; Step 2: Remove the average signal component before performing intrinsic orthogonal decomposition. The average value of a spatial discrete point based on the temperature snapshot matrix is expressed as: (21) Step 3: Use the singular value decomposition method to find the standard eigenvector v and the corresponding eigenvalue λ of the matrix. At this time, the eigenvalues are arranged from large to small. (22) Among them, V is the modal matrix, each column of which corresponds to a basis function; U is an M×M unitary matrix, S is an M×N non-negative real symmetric matrix, the elements of its diagonal are singular values, and the time coefficient matrix , where each column corresponds to the time constant of the mode; the modal energy is calculated by singular value, ; Step 4: Select the number of basis functions according to the energy contribution rate, take the first M eigenvalues and corresponding eigenvectors, and generate POD basis functions based on the eigenvalues and eigenvectors: (23) Step 5: Determine whether the approximate solution of the temperature field meets the accuracy requirements. If not, return to the first step of step 4 to reconstruct the full-order snapshot and then select the POD basis function until the error accuracy requirements are met. Step 5: Reconstruct the temperature field data based on the reduced-order model, including: Step 1: Based on the Galerkin method, the full-order temperature field data is subjected to low-dimensional linear combination by selecting the number of basis functions according to the energy contribution rate: (24) in, is the mode, that is, the basis function, a i (t) is the time constant of the corresponding mode; Step 2: Add the average value of step 2 in step 4 to reconstruct the temperature field data: (25) Step 3: Perform error analysis on the temperature field reconstructed by the data and the full-order temperature field obtained by the CFD method to minimize the error between the POD solution of the temperature field and the solution of the traditional numerical method: (26) in, is the numerical solution of the temperature field obtained by the numerical method, T(x,y,z) is the reduced-order solution of the temperature field obtained by the proper orthogonal decomposition method, Represents the square of the absolute value.

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