Conservative quantity conversion method in numerical simulation of multiphase leakage accident of fourth-generation reactor

Abstract

The invention discloses a conservation quantity transformation method in numerical simulation of a multi-phase leakage accident of a fourth-generation reactor. The conservation quantity transformation method comprises the following five steps: 1, determining a two-phase medium and working condition parameters; 2, describing a high-pressure-difference gas-liquid two-fluid model by adopting a conservation law mode, wherein the high-pressure-difference gas-liquid two-fluid model comprises a four-equation model, a five-equation model and a six-equation model; 3, determining an original variable expression and a conservation variable expression of the solving equation, and obtaining a numerical value of a conservation variable; 4, iteratively solving a nonlinear equation set according to a thermodynamic relationship of physical properties; and 5, calculating other original variables. The method has the advantages that the conservation characteristic of an equation set is not damaged, self-consistent physical property parameters and various original variables can be more stably and efficiently calculated, and the method can be easily applied to an existing tight template high-order method or other explicit conservation law solving methods; and a foundation is established for numerical simulation and safety evaluation software development of fourth-generation reactor multiphase leakage accidents.

Description

Technical Field

[0001] The present invention belongs to the technical field of nuclear energy development, and in particular relates to a conservation quantity transformation method in numerical simulation of a multiphase leakage accident in a fourth-generation reactor. Background Art

[0002] Fourth-generation advanced small nuclear reactors, represented by molten salt reactors and liquid metal-cooled fast reactors, utilize a highly efficient heat exchange system design involving a secondary supercritical fluid and a primary low-pressure coolant, with a pressure differential exceeding 10 MPa. However, in the design basis accident of a heat transfer tube rupture, a violent multiphase jet process occurs, resulting in effects such as static pressure wave shock, dynamic pressure inertia shock, and drastic changes in reactivity caused by bubble migration, seriously threatening the safety of the reactor core, vessel, and structural materials. Experimental studies on the safety analysis of high-pressure heat transfer tube rupture accidents in fourth-generation liquid-cooled reactors have insufficient resolution and are difficult to implement. Therefore, high-resolution and high-precision numerical simulation methods are needed to analyze local phenomena for more accurate safety assessments. These phenomena involve complex coupling issues such as compressible / weakly compressible mixing and multiphase and multiscale pressure waves, resulting in complex nonlinear effects and discontinuous solutions such as shock waves, which greatly increase the computational complexity and difficulty of numerical simulations.

[0003] High-precision methods hold promise for reducing computational complexity and easing simulation difficulty. These methods typically construct spatially high-order smooth solutions within the polynomial function space to compute spatial derivatives and employ high-order time integration in the time domain to obtain high-order solutions that are consistent across time and space. Compared to commonly used low-order finite volume and finite difference methods, these methods can exponentially reduce error while maintaining the same degrees of freedom. Consequently, they can also significantly reduce the number of degrees of freedom required for solving the problem while maintaining the same error, thereby saving computational resources. For complex geometric features in engineering problems, high-order schemes are often constructed using the finite volume method (FVM) or the discontinuous Galerkin method (DGM). High-order FVMs typically employ compact templates for reconstruction, but struggle to achieve high accuracy. High-order DGMs define the solution space within the polynomial space and can achieve arbitrary accuracy in the smooth solution region. In the discontinuous solution region, techniques such as limiters are used to achieve 4th-5th order accuracy comparable to mainstream high-order FVMs, yielding more accurate results. These high-precision numerical discretization methods can provide a theoretical foundation for the development of safety assessment software for analyzing and assessing the rupture of liquid-cooled reactor heat exchangers. However, a significant disadvantage of methods such as DGM is the large number of degrees of freedom. When meshing the heat transfer tube rupture conditions of liquid metal or molten salt reactors at a scale of about 300-500 mm, the number of sparse curved edge grids used is about 200,000. If fourth-order or higher polynomials are used, the total spatial degrees of freedom for solving the two-phase compressible Navier-Stokes (NS) equations reaches 100 million to 150 million or more. At this time, the cost of using implicit formats for time discretization and iterative solution is too high, so explicit formats are usually used for synchronous or asynchronous solution. In addition, in order to obtain better conservation, stability and correctly capture the shock wave position, the NS equations in the form of (hyperbolic) conservation laws are usually required.

[0004] However, using explicit formats and conservation law equations only allows for solving the system in the form of discretized ordinary differential equations. Therefore, after the time integration, the conserved variables at the next moment, such as mass, momentum, and total energy, are obtained, rather than the original variables, such as density, velocity, and temperature. For single-phase simulations, all the original variables can be directly calculated by substituting the conserved variables into the physical property functions. However, two-phase simulations present difficulties: it is difficult to explicitly convert the conserved variables back to the original variables, and the lack of such a conversion relationship can render the simulation impossible.

[0005] To solve this problem of converting conserved quantities into original variables, one approach is to introduce a new volume fraction transport equation. This increases the number of equations to be solved, and the volume fraction itself, as one of the conserved quantities, is updated as the time integral is completed. The original variables can then be calculated using the volume fraction and other conserved variables. However, this approach suffers from the difficulty of constraining the correct physical property relationships, and the other original variables calculated may contradict each other. This type of problem is described in more detail in Abgrall's theorem. Furthermore, the non-conserved volume fraction transport equation and the conserved continuity equation must be solved simultaneously to satisfy the Abgrall condition. Therefore, this approach introduces more computational cost and complexity to the DGM. Summary of the Invention

[0006] In order to overcome the problems existing in the above-mentioned prior art, the present invention proposes a method for transforming conserved quantities in the numerical simulation of multiphase leakage accidents in fourth-generation reactors. This method does not destroy the conservation characteristics of the equation group and does not increase the degrees of freedom of the solution. It can calculate self-consistent physical parameters and various original variables more stably and efficiently, and is easy to apply in the numerical simulation of multiphase leakage accidents in fourth-generation reactors.

[0007] The conservation quantity transformation method in the numerical simulation of the fourth generation reactor multiphase leakage accident includes the following steps:

[0008] Step 1. Determine the two-phase medium and operating parameters: the liquid working medium is limited to a weakly compressible fluid, its sound velocity should be higher than 1000m / s, the operating temperature is between 100-700℃, and the operating pressure is between 0.1-30MPa; the gas working medium is limited to the heat transfer working medium of the power cycle system, its sound velocity is higher than 150m / s in the full operating range, the operating temperature is between 100-700℃, and the operating pressure is between 0.1-30MPa;

[0009] Step 2: Use conservation laws to describe the high-pressure differential gas-liquid two-fluid model: During a multiphase leakage accident, a two-phase conservation law equation system based on ensemble average theory is used to describe the physical process. These two-phase conservation law equations include four-equation models, five-equation models, and six-equation models. These models use mass, momentum, energy, and volume fraction as conservation dependent variables, and velocity, temperature, pressure, and volume fraction as dependent variables.

[0010] The Euler two-fluid four-equation model of conservation law in the multiphase leakage accident process is as follows:

[0011]

[0012] The Euler two-fluid five-equation model of conservation law in the multiphase leakage accident process is as follows:

[0013]

[0014] The Euler two-fluid six-equation model of conservation law in the multiphase leakage accident process is as follows:

[0015]

[0016] In formulas (1), (2) and (3), t represents time; α f represents the liquid volume fraction; α g represents the gas phase volume fraction; ρ f represents the liquid density; ρ g represents the gas phase density; v represents the mixing velocity of the two phases of the fluid; v f represents the liquid phase fluid velocity; v g represents the velocity of gas phase fluid; e f represents the total energy of the liquid; e g represents the total energy of the gas; Γ g is the interphase mass transfer term; p is the static pressure; I is the second-order unit tensor; K is the viscous force tensor; Υ is the interphase force; q is the local heat flux density; Θ is the interphase heat transfer power; is the divergence operator; all variables are in SI units;

[0017] Step 3: Determine the expression of the conserved variable to solve the equation:

[0018] During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid four-equation model of conservation law type is:

[0019]

[0020] In formula (4), F c，f Flux represents the mass of the liquid phase, which is a conserved variable; c，f represents the liquid mass flux function vector; S c，f represents the source term of the liquid mass equation; F c，g Represents the gas phase mass, as a conserved variable; Flux c，g represents the gas phase mass flux function vector; S c，g represents the source term of the gas phase mass equation; F m Represents the two-phase mixed momentum vector as a conserved variable vector; Flux m represents the two-phase momentum flux function tensor; S m represents the source term vector of the two-phase momentum equation; F e Flux represents the two-phase mixing energy as a conserved variable; e represents the two-phase mixing energy flux function vector; S e represents the source term of the two-phase mixture energy equation; all variables are in SI units;

[0021] During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid five-equation model of conservation law type is:

[0022]

[0023] In formula (5), F e,f Flux represents the liquid phase energy as a conserved variable; e,f represents the liquid phase energy flux function vector; S e,f represents the source term of the liquid phase energy equation; F e，g Flux represents the gas phase energy as a conserved variable; e，g represents the gas phase energy flux function vector; S e，g represents the source term of the gas phase energy equation; all variables are in international standard units;

[0024] During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid six-equation model of conservation law type is:

[0025]

[0026] In formula (6), F m.f Flux represents the liquid phase momentum vector, which is a conserved variable vector; m,f represents the liquid phase momentum flux function tensor; S m，f represents the source term vector of the liquid phase momentum equation; F m.g represents the gas phase momentum vector, which is a conserved variable vector; Flux m，g represents the gas phase momentum flux function tensor; S m，g represents the source term vector of the liquid phase momentum equation; all variables are in SI units;

[0027] Step 4: Iteratively solve the nonlinear equations based on the thermodynamic relationship of physical properties:

[0028] It is necessary to determine the conversion relationship between the conservation variables in the two-phase conservation law equations of the ensemble average theory used in the reactor multiphase leakage accident and the original variables, and then use an iterative method to convert the conservation variables at each time step in the numerical solution process into the original variables;

[0029] For the four-equation model, the solution process is abstracted into the following solution process:

[0030]

[0031] In formula (7), T represents temperature; f1 and f2 represent nonlinear equations; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function at temperature T; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function at temperature T; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function for static pressure p; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function for the static pressure p; for an immiscible two-phase system, substitute the respective state equations and Solve the simultaneous nonlinear equations using Newton's method to obtain (p, T) and calculate other state variables. For boiling systems, replace the input variables (p, T) with (p), simplify the equations f1(p, T) and f2(p, T) into a single equation f1(p), and use the bisection method to solve the equation and calculate other state variables.

[0032] For the five-equation model, the solution process is abstracted into the following solution process:

[0033]

[0034] In formula (8), u f represents the internal energy of the liquid phase; u g represents the internal energy of the gas phase; F m，x represents the spatial x-direction component of the two-phase mixed momentum; F m,y represents the spatial y-direction component of the two-phase mixed momentum; F m,z represents the spatial z-direction component of the two-phase mixed momentum; v x represents the spatial x-direction component of the two-phase mixing velocity; v y represents the spatial y-direction component of the two-phase mixing velocity; v z Represents the spatial z-direction component of the two-phase mixing velocity; Expressed by the liquid internal energy u f 、Calculate the liquid density ρ using the static pressure p f Any state equation function; Expressed by the gas phase internal energy u g , static pressure p to calculate gas density ρ g Any state equation function; f(p) represents a single nonlinear equation with static pressure p as the independent variable; R1 represents the first iteration value of the gas phase volume fraction; R2 represents the second iteration value of the gas phase volume fraction; all variables are in international standard units; for the two insoluble phases, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two phases that are miscible, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables;

[0035] For the six-equation model, the solution process is abstracted into the following solution process:

[0036]

[0037] In formula (9), F m，f，x represents the spatial x-direction component of the liquid phase momentum; F m，f，y represents the spatial y-direction component of the liquid phase momentum; F m，f，z represents the spatial z-direction component of the liquid phase momentum; F m，g，x represents the spatial x-direction component of the gas phase momentum; F m，g，y represents the spatial y-direction component of the gas phase momentum; F m，g，z represents the spatial z-direction component of the gas phase momentum; v f，x represents the spatial x-direction component of the liquid phase velocity; v f，y represents the spatial y-direction component of the liquid phase velocity; v f，z Represents the spatial z-direction component of the liquid phase velocity; v g，x represents the spatial x-direction component of the gas phase velocity; v g，y represents the spatial y-direction component of the gas phase velocity; v g，z represents the spatial z-direction component of the gas phase velocity; all variables are in SI units; for the two insoluble phases, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two phases that are miscible, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables;

[0038] Step 5. Calculate the remaining original variables:

[0039] For the four-equation, five-equation, and six-equation models of the Euler two-fluid model of the conservation law type in the process of the multiphase leakage accident mentioned above, after solving the equation group, the pressure p or pressure temperature (p, T) is substituted into (7), (8), or (9), and the other original variables are calculated to form an ordinary differential equation system to ensure the smooth progress of the solution.

[0040] Preferably, the weakly compressible fluid is sodium, molten salt, or a lead-based alloy, which is notably characterized by its high acoustic velocity and density. The conservation quantity transformation method of the present invention ensures the self-consistency of the physical properties of the weakly compressible fluid during numerical simulations. The binary equation of state, which relates density to temperature and pressure, accurately describes the propagation behavior of pressure waves in the liquid. This is unattainable with traditional weakly compressible simulations that are independent of density and pressure.

[0041] Preferably, the heat transfer medium of the power cycle system is water vapor, carbon dioxide, or helium, and its notable feature is a physical property equation of state with binary parameters. The conservation quantity transformation method of the present invention accurately describes high-Mach number phenomena such as normal shock waves, oblique shock waves, and expansion waves that occur in the gas, while ensuring self-consistency between physical properties and volume fractions.

[0042] Compared with the prior art, the present invention has the following advantages:

[0043] The present invention proposes a method for transforming conservation variables in the numerical simulation of multiphase leakage accidents in fourth-generation reactors, which is applied to nuclear power systems of fourth-generation reactors such as molten salt or liquid metal reactors with high-pressure power conversion system working fluids; when a design benchmark accident of heat transfer tube rupture occurs, it is used to finely simulate and analyze the accident process, improve the conversion problem of conservation variables and original variables in the two-phase flow explicit high-precision numerical simulation method in the accident simulation, and provide a basis for refining the safety review criteria. For the case where a single equation is used in the conversion process, the present invention adopts the bisection method to find the root; for double equations, the efficient Newton method is used for calculation. Compared with the existing method of adding additional volume fraction transport equations, the method of the present invention does not destroy the conservation characteristics of the equation group and does not increase the degree of freedom of the solution. It can more stably and efficiently calculate self-consistent physical parameters and various original variables, and is easy to apply in the numerical simulation of multiphase leakage accidents in fourth-generation reactors. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is a schematic diagram of the conservation quantity transformation method in the numerical simulation of multiphase leakage accidents in fourth-generation reactors. DETAILED DESCRIPTION

[0045] The present invention will be described in detail below with reference to the accompanying drawings and embodiments:

[0046] like Figure 1 As shown, the present invention proposes a method for transforming conservation variables in the numerical simulation of multiphase leakage accidents in fourth-generation reactors, which includes the following five steps: 1. determining the two-phase medium and operating parameters; 2. describing the high-pressure differential gas-liquid two-fluid model using conservation laws; 3. determining the conservation variable expressions for solving the equations; 4. iteratively solving the nonlinear equation (group) based on the thermodynamic relationship of physical properties; 5. calculating the remaining original variables;

[0047] Step 1. Determine the two-phase medium and operating parameters: the liquid working medium is limited to weakly compressible fluids, including but not limited to sodium, molten salt, and lead-based alloys, and its sound velocity should generally be higher than 1000m / s, the operating temperature is between 100-700°C, and the operating pressure is between 0.1-30MPa; the gas working medium is limited to commonly used heat transfer working fluids in power cycle systems, including but not limited to water vapor, carbon dioxide, and helium, and its sound velocity is higher than 150m / s in the full operating range, the operating temperature is between 100-700°C, and the operating pressure is between 0.1-30MPa;

[0048] Step 2: Use conservation laws to describe the high-pressure differential gas-liquid two-fluid model: During a multiphase leakage accident, a two-phase conservation law equation system based on ensemble average theory is used to describe the physical process. The two-phase conservation law equations include four-equation models, five-equation models, and six-equation models, with mass, momentum, energy, and volume fraction as conservation dependent variables, and original variables such as velocity, temperature, pressure, and volume fraction as dependent variables.

[0049] The Euler two-fluid four-equation model of conservation law is as follows:

[0050]

[0051] The Euler two-fluid five-equation model of conservation law is as follows:

[0052]

[0053] The Euler two-fluid six-equation model of conservation law is as follows:

[0054]

[0055] In formulas (1), (2) and (3), t represents time; α f represents the liquid volume fraction; α g represents the gas phase volume fraction; ρ f represents the liquid density; ρ g represents the gas phase density; v represents the mixing velocity of the two phases of the fluid; v f represents the liquid phase fluid velocity; v g represents the velocity of gas phase fluid; e f represents the total energy of the liquid; e g represents the total energy of the gas; Γ g is the interphase mass transfer term; p is the static pressure; I is the second-order unit tensor; K is the viscous force tensor; Υ is the interphase force; q is the local heat flux density; Θ is the interphase heat transfer power; is the divergence operator; all variables are in SI units;

[0056] Step 3: Determine the expression of the conserved variable to solve the equation:

[0057] The conservation variable expression of the four-equation model is:

[0058]

[0059] In formula (4), F c，f Flux represents the mass of the liquid phase, which is a conserved variable; c，f represents the liquid mass flux function vector; S c，f represents the source term of the liquid mass equation; F c，g Represents the gas phase mass, as a conserved variable; Flux c，g represents the gas phase mass flux function vector; S c，g represents the source term of the gas phase mass equation; F m Represents the two-phase mixed momentum vector as a conserved variable vector; Flux m represents the two-phase momentum flux function tensor; S m represents the source term vector of the two-phase momentum equation; F e Flux represents the two-phase mixing energy as a conserved variable; e represents the two-phase mixing energy flux function vector; S e represents the source term of the two-phase mixture energy equation; all variables are in SI units;

[0060] The conservation variable expression of the five-equation model is:

[0061]

[0062] In formula (5), F e，f Flux represents the liquid phase energy as a conserved variable; e，f represents the liquid phase energy flux function vector; S e，f represents the source term of the liquid phase energy equation; F e，g Flux represents the gas phase energy as a conserved variable; e，g represents the gas phase energy flux function vector; S e，g represents the source term of the gas phase energy equation; all variables are in international standard units;

[0063] The conservation variable expression of the six-equation model is:

[0064]

[0065] In formula (6), F m.f Flux represents the liquid phase momentum vector, which is a conserved variable vector; m,f represents the liquid phase momentum flux function tensor; S m，f represents the source term vector of the liquid phase momentum equation; F m.g represents the gas phase momentum vector, which is a conserved variable vector; Flux m，grepresents the gas phase momentum flux function tensor; S m，g represents the source term vector of the liquid phase momentum equation; all variables are in SI units;

[0066] Step 4: Iteratively solve the nonlinear equation (set) based on the thermodynamic relationship of physical properties:

[0067] It is necessary to determine the conversion relationship between the conservation dependent variables and the original dependent variables in the two-phase conservation law equations (four equations, five equations, and six equations) used in the ensemble average theory during the reactor multiphase leakage accident, and then use an iterative method to convert the conservation variables at each time step in the numerical solution process into the original variables;

[0068] For the four-equation model, the solution process is abstracted into the following solution process:

[0069]

[0070] In formula (7), T represents temperature; f1 and f2 represent nonlinear equations; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function at temperature T; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function at temperature T; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function for static pressure p; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function for static pressure p; for insoluble two-phase systems, such as lead bismuth-carbon dioxide, substitute their respective state equations and Solve the simultaneous nonlinear equations using Newton's method to obtain (p, T), which can then be used to calculate other state variables. For boiling systems, such as water-steam, replace the input variables (p, T) with (p), simplifying the equations f1(p, T) and f2(p, T) into a single equation f1(p), and use the bisection method to solve the equation and calculate other state variables.

[0071] Let's calibrate the solver:

[0072] Taking the boiling system as an example, a two-phase equilibrium state is given, such as the thermodynamic gas content x = 0.5, the temperature is T = 100 ° C, and the saturation pressure is p = p (T, x). Let the cross-sectional gas content α g = 0.4, velocity field v = [1, 2, 3] TCalculate the conserved variables F1 and F2, set the initial pressure value, and use the Brent secant bisection method to solve the nonlinear equations. Table 1 shows the error between the variable conversion calculation and the machine calculation results for the four-equation boiling system. The error of the original variables is close to the machine accuracy, proving that the solution is effective.

[0073] Table 1: Errors in variable conversion calculation results for the four-equation model boiling system

[0074] Static pressure error -1.5783388882678327e-15 Volume fraction error 1.3877787887814457e-15 Liquid density error 4.833345132511662e-15 Gas phase density error -1.4848266021806921e-15 Liquid internal energy error -1.8834233518838957e-14 Gas phase internal energy error -5.5745059893437762-16

[0075] Taking an insoluble system as an example, a two-phase equilibrium state is given, where air and carbon dioxide are mixed, with air as phase 1 and carbon dioxide as phase 2. The temperature is T = 500°C and the pressure is p = 0.1 MPa. Let the cross-sectional air content α be g = 0.4, velocity field v = [1, 2, 3] T The conserved variables F1, F2, F3, and F4 were calculated, initial values ​​for temperature and pressure were set, and the nonlinear system of equations was solved using the Newton-Krylov method constructed using the numerical Jacobi. Table 2 shows the error comparison between the variable conversion calculation and the machine calculation results for the four-equation model in an insoluble system. The error in the original variables is close to the machine accuracy, demonstrating the effectiveness of the solution.

[0076] Table 2: Variable conversion calculation results and accuracy of the four-equation model for insoluble systems

[0077] Static pressure error 0.0 Volume fraction error 1.3877787807814457e-16 Liquid density error -1.6217694882692622e-16 Gas phase density error -1.2324865796694746e-16 Liquid phase internal energy error 0.0 Gas phase internal energy error 0.0

[0078] For the five-equation model, the solution process is abstracted into the following solution process:

[0079]

[0080] In formula (8), u represents internal energy; subscripts x, y, and z represent three orthogonal coordinate directions; f represents a single nonlinear equation; the meanings of the remaining variables are the same as those in formulas (1), (2), and (3); all variables are in international standard units; for insoluble two-phase, such as lead-bismuth-carbon dioxide, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two mutually soluble phases, such as water-water vapor, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables;

[0081] Let's calibrate the solver:

[0082] Taking the boiling system as an example, a two-phase equilibrium state is given, such as the thermodynamic gas content x = 0.5, the temperature is T = 100 ° C, and the saturation pressure is p = p (T, x). Let the cross-sectional gas content α g =0.4. Speed ​​[v x v y v z ] T =[1 2 3] T Calculate the conserved variables F1–F5, set the initial pressure value, and use the Brent secant bisection method to solve the nonlinear equations, thereby calculating the original variables. Table 3 shows the error comparison between the variable conversion calculation and the machine calculation results for the boiling system using the five-equation model. The highest error in the original variables is less than 1e3, representing the internal energy error; the remaining errors are all on the order of 1e-5. While the solution is valid, it is not as accurate as the four-equation model.

[0083] Table 3: Variable conversion calculation results and accuracy for the five-equation model boiling system

[0084] Static pressure error -7.13545885121403e-5 Volume fraction error 2.429812927376429e-5 Liquid density error 1.619931525681478e-5 Gas phase density error -2.4797538889879666e-5 Liquid internal energy error -0.00021715258832298817 Gas phase internal energy error -3.631252835538793e-5

[0085] For the six-equation model, the solution process is abstracted into the following solution process:

[0086]

[0087] The meanings of the variables in formula (9) are the same as those in formulas (1), (2) and (3); all variables are in international standard units; for insoluble two phases, such as lead bismuth-carbon dioxide, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two mutually soluble phases, such as water-water vapor, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables;

[0088] Let's calibrate the solver:

[0089] Taking the boiling system as an example, a two-phase equilibrium state is given, such as the thermodynamic gas content x = 0.5, the temperature is T = 100 ° C, and the saturation pressure is p = p (T, x). Let the cross-sectional gas content α g =0.4. Liquid velocity [v x，f v y，f v z，f ] T =[0.1 0.20.3] T , gas velocity [v x，g v y，g v z，g ] T =[1 2 3]T Calculate the conserved variables F1 to F6; set the initial pressure value, solve the nonlinear equations using the Brent secant bisection method, and then calculate the original variables. Table 4 shows the results of the variable conversion calculation for the boiling system of the six-equation model. The maximum error of the original variables is less than 1e-4, indicating that the solution is valid. The accuracy is higher than that of the five-equation model but lower than that of the four-equation model.

[0090] Table 4: Variable conversion calculation results and accuracy for the six-equation model boiling system

[0091] Static pressure error 1.8784478674516235e-7 Volume fraction error -4.37555883243717e-5 Liquid density error -2.916949818961345e-5 Gas phase density error 4.375741495191123e-5 Liquid phase internal energy error 1.436547310888124e-7 Gas phase internal energy error -3.631252885538793e-5

[0092] Step 5. Calculate the remaining original variables:

[0093] For the four-, five-, and six-equation models above, after solving the equations, the pressure p or pressure-temperature (p, T) can be substituted into (7), (8), or (9) to calculate the other original variables. The calculation errors are summarized in step 4.

[0094] The above content is a further detailed description of the present invention in combination with specific preferred embodiments. It cannot be considered that the specific embodiments of the present invention are limited to these. For ordinary technicians in the technical field to which the present invention belongs, they can make several simple deductions or substitutions without departing from the concept of the present invention, which should be regarded as belonging to the scope of patent protection determined by the submitted claims of the present invention.

Claims

1. A method for transforming conservation quantities in numerical simulation of multiphase leakage accidents in fourth-generation reactors, characterized by: The steps include: Step 1. Determine the two-phase medium and operating parameters: the liquid working medium is limited to a weakly compressible fluid, its sound velocity should be higher than 1000m / s, the operating temperature is between 100-700℃, and the operating pressure is between 0.1-30MPa; the gas working medium is limited to the heat transfer working medium of the power cycle system, its sound velocity is higher than 150m / s in the full operating range, the operating temperature is between 100-700℃, and the operating pressure is between 0.1-30MPa; Step 2: Use conservation laws to describe the high-pressure differential gas-liquid two-fluid model: During a multiphase leakage accident, a two-phase conservation law equation system based on ensemble average theory is used to describe the physical process. These two-phase conservation law equations include four-equation models, five-equation models, and six-equation models. These models use mass, momentum, energy, and volume fraction as conservation dependent variables, and velocity, temperature, pressure, and volume fraction as dependent variables. The Euler two-fluid four-equation model of conservation law in the multiphase leakage accident process is as follows: The Euler two-fluid five-equation model of conservation law in the multiphase leakage accident process is as follows: The Euler two-fluid six-equation model of conservation law in the multiphase leakage accident process is as follows: In formulas (1), (2) and (3), t represents time; α f represents the liquid volume fraction; α g represents the gas phase volume fraction; ρ f represents the liquid density; ρ g represents the gas phase density; v represents the mixing velocity of the two phases of the fluid; v f represents the liquid phase fluid velocity; v g represents the velocity of gas phase fluid; e f represents the total energy of the liquid; e g represents the total energy of the gas; Γ g is the interphase mass transfer term; p is the static pressure; I is the second-order unit tensor; K is the viscous force tensor; Υ is the interphase force; q is the local heat flux density; Θ is the interphase heat transfer power; is the divergence operator; all variables are in SI units; Step 3: Determine the expression of the conserved variable to solve the equation: During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid four-equation model of conservation law type is: In formula (4), F c，f Flux represents the mass of the liquid phase, which is a conserved variable; c，f represents the liquid mass flux function vector; S c，f represents the source term of the liquid mass equation; F c，g Represents the gas phase mass, as a conserved variable; Flux c，g represents the gas phase mass flux function vector; S c，g represents the source term of the gas phase mass equation; F m Represents the two-phase mixed momentum vector as a conserved variable vector; Flux m represents the two-phase momentum flux function tensor; S m represents the source term vector of the two-phase momentum equation; F e Flux represents the two-phase mixing energy as a conserved variable; e represents the two-phase mixing energy flux function vector; S e represents the source term of the two-phase mixture energy equation; all variables are in SI units; During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid five-equation model of conservation law type is: In formula (5), F e，f Flux represents the liquid phase energy as a conserved variable; e，f represents the liquid phase energy flux function vector; S e，f represents the source term of the liquid phase energy equation; F e，g Flux represents the gas phase energy as a conserved variable; e，g represents the gas phase energy flux function vector; S e，g represents the source term of the gas phase energy equation; all variables are in international standard units; During a multiphase leakage accident, the conservation variable expression of the Euler two-fluid six-equation model of conservation law type is: In formula (6), F m.f Flux represents the liquid phase momentum vector, which is a conserved variable vector; m，f represents the liquid phase momentum flux function tensor; S m，f represents the source term vector of the liquid phase momentum equation; F m.g represents the gas phase momentum vector, which is a conserved variable vector; Flux m，g represents the gas phase momentum flux function tensor; S m，g represents the source term vector of the liquid phase momentum equation; all variables are in SI units; Step 4: Iteratively solve the nonlinear equations based on the thermodynamic relationship of physical properties: It is necessary to determine the conversion relationship between the conservation variables in the two-phase conservation law equations of the ensemble average theory used in the reactor multiphase leakage accident and the original variables, and then use an iterative method to convert the conservation variables at each time step in the numerical solution process into the original variables; For the four-equation model, the solution process is abstracted into the following solution process: In formula (7), T represents temperature; f1 and f2 represent nonlinear equations; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function at temperature T; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function at temperature T; Expressed by the liquid density ρ f , total liquid energy e f Calculate any state equation function for static pressure p; Expressed by the gas phase density ρ g , gas phase total energy e g Calculate any state equation function for the static pressure p; for an immiscible two-phase system, substitute the respective state equations and Solve the simultaneous nonlinear equations using Newton's method to obtain (p, T) and calculate other state variables. For boiling systems, replace the input variables (p, T) with (p), simplify the equations f1(p, T) and f2(p, T) into a single equation f1(p), and use the bisection method to solve the equation and calculate other state variables. For the five-equation model, the solution process is abstracted into the following solution process: In formula (8), u f represents the internal energy of the liquid phase; u g represents the internal energy of the gas phase; F m，x represents the spatial x-direction component of the two-phase mixed momentum; F m，y represents the spatial y-direction component of the two-phase mixed momentum; F m，z represents the spatial z-direction component of the two-phase mixed momentum; v x represents the spatial x-direction component of the two-phase mixing velocity; v y represents the spatial y-direction component of the two-phase mixing velocity; v z Represents the spatial z-direction component of the two-phase mixing velocity; Expressed by the liquid internal energy u f 、Calculate the liquid density ρ using the static pressure p f Any state equation function; Expressed by the gas phase internal energy u g , static pressure p to calculate gas density ρ g Any state equation function; f(p) represents a single nonlinear equation with static pressure p as the independent variable; R1 represents the first iteration value of the gas phase volume fraction; R2 represents the second iteration value of the gas phase volume fraction; all variables are in international standard units; for the two insoluble phases, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two phases that are miscible, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables; For the six-equation model, the solution process is abstracted into the following solution process: In formula (9), F m，f，x represents the spatial x-direction component of the liquid phase momentum; F m，f，y represents the spatial y-direction component of the liquid phase momentum; F m，f，z represents the spatial z-direction component of the liquid phase momentum; F m，g，x represents the spatial x-direction component of the gas phase momentum; F m，g，y Represents the spatial y-direction component of the gas phase momentum; F m，g，z represents the spatial z-direction component of the gas phase momentum; v f，x represents the spatial x-direction component of the liquid phase velocity; v f，y represents the spatial y-direction component of the liquid phase velocity; v f，z Represents the spatial z-direction component of the liquid phase velocity; v g，x represents the spatial x-direction component of the gas phase velocity; v g，y represents the spatial y-direction component of the gas phase velocity; v g，z represents the spatial z-direction component of the gas phase velocity; all variables are in SI units; for the two insoluble phases, and Substitute the respective physical state equations for calculation, use the dichotomy method to solve the equations and calculate other state variables; for two phases that are miscible, and Substitute the same state equation, use the bisection method to solve the equation and calculate other state variables; Step 5. Calculate the remaining original variables: For the four-equation, five-equation, and six-equation models of the Euler two-fluid model of the conservation law type in the process of the multiphase leakage accident mentioned above, after solving the equation group, the pressure p or pressure temperature (p, T) is substituted into (7), (8), or (9), and the other original variables are calculated to form an ordinary differential equation system to ensure the smooth progress of the solution.

2. The method for transforming conserved quantities in numerical simulation of multiphase leakage accidents in fourth-generation reactors according to claim 1, characterized in that: The weakly compressible fluid is sodium, molten salt or lead-based alloy.

3. The method for transforming conserved quantities in numerical simulation of multiphase leakage accidents in fourth-generation reactors according to claim 1, characterized in that: The heat transfer medium of the power cycle system is water vapor, carbon dioxide or helium.

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