A finite element analysis method for quench process of superconducting magnet

Abstract

The application discloses a kind of finite element analysis methods of superconducting magnet quench process.The method first obtains the change of current in coil with time in quench process, then establishes electromagnetic field control equation and heat conduction equation, by sequentially solving two equations, the eddy current density, electromagnetic force and temperature distribution in magnet at each time are obtained.The method fully considers the coupling of electromagnetic field and temperature field, can accurately simulate the change of eddy current density, electromagnetic force, temperature with time after magnet quench.The application has high simulation precision, can provide effective guidance for reasonable design of magnet structure.

Description

TECHNICAL FIELD

[0001] The present application relates to a finite element analysis method, more particularly to a finite element analysis method for quench of superconducting magnet. BACKGROUND

[0002] Quench of superconducting magnet is a very important problem, and is also an inherent problem of superconducting magnet itself. If the process of generating superconducting magnet is not mature, the quench times of each magnet can reach more than 5 times. Even in the case of mature process, sometimes artificial quench is needed to judge the stability of the magnet. In the process of quench, the electromagnetic energy stored in the coil is fully released, thus a large amount of heat is generated to make liquid helium evaporate, thereby causing serious loss. In addition, a large electromagnetic force is generated on the metal wall inside the magnet, so that the cold shield, end cover, skeleton and the like produce elastic strain. When the structure of the magnet is not reasonably designed, quench will cause damage to the components inside the magnet, thereby affecting the stability of the magnet. In severe cases, some metal components can even produce plastic strain. Therefore, it is necessary to analyze the quench process of the magnet. At present, the analysis of the quench process of the magnet coil is mainly focused on the magnet coil itself, and few documents consider the influence of quench on the surrounding conductors such as the cold shield. According to the foregoing, it is undoubtedly necessary to analyze the stress and temperature distribution of the metal components such as the cold shield in the quench process. However, the parameters such as the electrical conductivity of the conductor will change with the rise of temperature, which undoubtedly increases the difficulty of accurate simulation. SUMMARY

[0003] The present application aims at providing an effective method for analyzing the quench process of superconducting magnet by using time domain finite element analysis. By using the method, the electric field, magnetic field, eddy current density and electromagnetic force generated on the conductor during the quench process of the superconducting coil can be analyzed, thereby providing guidance for reasonably designing the structure of the magnet.

[0004] Technical scheme: In order to achieve the above-mentioned application purpose, the present application provides a finite element analysis method for quench process of superconducting magnet. First, the relationship curve I(t) of the current in the magnet coil changing with time after the magnet quenches is obtained. Then, the initial temperature of each part of the magnet before quench is determined by taking I(t) as the excitation source and taking the coordinate at the quench starting time as t=0. The eddy current density generated on the conductor in the magnet at t=t1 is calculated by using time domain numerical simulation algorithm. Then, the heat generated by the eddy current density at any point on the conductor in the magnet system is calculated by the following formula

[0005]

[0006] In the above formula, is the coordinate point​ conductivity at the point, is the modulus of the vortex density at the point;

[0007] The following heat conduction control equation is then solved to obtain the temperature distribution in the conductor at t = t1:

[0008]

[0009] In the above equation, T is temperature, p is material density, c is specific heat of the material, x, y, z are coordinates in the Cartesian coordinate system, k x ,k y ,k z is the thermal conductivity of the metal at the point

[0010] The conductivity of the conductor at each coordinate point is then re-determined according to the temperature distribution, and the above process is repeated to calculate the vortex density and temperature at the next time until the calculation is completed.

[0011] Further, the current-time curve after the coil loses superconductivity in step (1) is obtained by sampling the current signal through the magnet monitoring device or by an ellipsoid diffusion model.

[0012] Further, the vortex density at each time is numerically simulated using the finite element method, and the following electromagnetic field control equation is used for finite element modeling:

[0013] In the conductor region:

[0014]

[0015] In the non-conductor region:

[0016]

[0017] In the above equation, A is the vector magnetic potential generated by the vortex density, V is the scalar potential, v is the inverse of the magnetic permeability, is the conductivity of the metal at the point , J s is the current density in the coil, and the symbol represents the curl operation, represents the divergence operation, represents the gradient operation.

[0018] Further, the vortex density in the conductor is obtained by the following formula:

[0019]

[0020] Further, when the maximum vortex density in the conductor satisfies ​When the simulation ends, epsilon is the set threshold value.

[0021] Further, for the heat conduction control equation, a tetrahedral mesh is adopted for dissection, and a node-based function is adopted as the interpolation function of the temperature field.

[0022] Further, a tetrahedral mesh is adopted for dissection, and an edge-based function is adopted as the interpolation function of the magnetic vector potential.

[0023] Further, the Lorentz force density at any point on the conductor in the magnet system is calculated by the following formula:

[0024]

[0025] In the above formula, is the vortex density at the coordinate point , and is the magnetic field at the coordinate point .

[0026] Beneficial effects: The analysis method has the advantages that: a time-domain finite element method for analyzing the quenching process is provided, the coupling between the electromagnetic field and the thermal field is accurately considered, the vortex density, the Lorentz force, and the heat and other parameter changes after the magnet quenches can be accurately obtained, and the simulation precision is very high. BRIEF DESCRIPTION OF DRAWINGS

[0027] Figure 1 is the method flowchart of the embodiment of the application.

[0028] Figure 2 is the current-time curve diagram of the magnet after quenching in the embodiment of the application.

[0029] Figure 3 is the vortex density curve diagram along the axial direction on the cold shield after quenching for 1.2s in the embodiment of the application.

[0030] Figure 4 is the Lorentz force density curve diagram along the axial direction on the cold shield after quenching for 1.2s in the embodiment of the application. DETAILED DESCRIPTION

[0031] The application relates to a method for analyzing the quenching problem of a superconducting magnet. The application discloses a scheme for analyzing the quenching process of a superconducting magnet by using a time-domain finite element method.

[0032] As Figure 1The application discloses a finite element analysis method for a quench process of a superconducting magnet. First, a current-time curve I(t) of a magnet coil after the magnet quenches is obtained; then, the I(t) is taken as an excitation source, a coordinate at a quench starting moment is defined as t=0, initial temperatures of each part in the magnet before the quench are determined, and an eddy current density generated on a conductor in the magnet at t=t1 is calculated by using a time-domain numerical simulation algorithm; then, heat generated by the eddy current density at any point of the conductor in the magnet system is calculated by the following formula

[0033]

[0034] In the formula, σ is an electrical conductivity of the point , J is a module value of the eddy current density of the point .

[0035] Then, a temperature distribution of the conductor at t=t1 is obtained by solving the following heat conduction control equation:

[0036]

[0037] In the formula, T is temperature, p is material density, c is material specific heat, x, y and z are coordinates in a Cartesian coordinate system, k x ,k y ,k z are thermal conductivities of the point in x, y and z directions; the directions of the three coordinate axes can be determined according to convenience of analysis problems. For the superconducting magnet problem, a central axis of the magnet is defined as the z axis, and any two orthogonal directions in a plane perpendicular to the z axis are defined as the x axis and the y axis.

[0038] The electrical conductivity of the conductor at each coordinate point is determined again according to the temperature distribution, and then the above process is repeated to calculate the eddy current density and the temperature at the next moment until the calculation is completed.

[0039] Before the method is adopted, the current-time curve after the coil quenches needs to be obtained. For a prepared magnet, the curve can be obtained by collecting the current in the coil after the magnet quenches. In the magnet design stage, an ellipsoid diffusion model can be used to calculate the current-time curve after the coil quenches. The model is described in some documents, such as Superconducting Magnet Design Basis by Nan Heli, and the specific calculation method is not given here.

[0040] ​​After the current in the coil is obtained, the eddy current density in the magnet at each time is simulated by using the finite element method. Different control equations can be used in the finite element eddy current density simulation. In this embodiment, the following electromagnetic field control equation is used for finite element modeling:

[0041] In the conductor region:

[0042]

[0043] In the non-conductor region:

[0044]

[0045] In the above formula, A is the vector magnetic potential generated by the eddy current density, V is the scalar potential, v is the inverse of the magnetic permeability, is the coordinate point of the metal, J s is the current density in the coil, and the symbol represents the curl operation, represents the divergence operation, represents the gradient operation.

[0046] Through analysis, it can be obtained that the above control equation has a unique solution after the boundary condition is determined. After the vector magnetic potential and the scalar potential in the magnet are obtained, the eddy current density in the conductor is obtained by the following formula:

[0047]

[0048] After the current density is obtained, the Lorentz force density at any point on the conductor in the magnet system is calculated by the following formula:

[0049]

[0050] In the above formula, is the coordinate point of the eddy current density, is the coordinate point of the magnetic field.

[0051] In this way, through the above simulation, we can obtain the eddy current density, electromagnetic force, and temperature at any time in the magnet. Because the electrical conductivity of the conductor changes with temperature at low temperature, we need to solve the electromagnetic field equation and the temperature equation during simulation, and update the value of the electrical conductivity after solving to ensure the accuracy of the simulation.

[0052] In the above algorithm, the exit criterion of time iteration needs to be set. We can use the maximum eddy current density value as the criterion, and the simulation ends when the maximum eddy current density value in the conductor satisfies , where ε is a value set artificially.

[0053] For the heat conduction equation, we use tetrahedron mesh to divide the domain, and use node-based function as the interpolation function of temperature field. For the electromagnetic control equation, we also use tetrahedron mesh to divide the domain, and use edge-based function as the interpolation function of magnetic vector potential. After the specific form of control equation and mesh, basis function are determined, we can use the Galerkin method to disperse the control equation. The steps of using the Galerkin method to disperse the control equation are described in some literature, such as Jin Jianming's Electromagnetic Field Finite Element Method, which will not be repeated here.

[0054] A specific simulation example is given below. Given the current amplitude changes with time obtained by monitoring after the superconducting magnet loses superconductivity as shown in Figure 2 In order to optimize the magnet cold shield structure, it is necessary to simulate the loss of superconductivity process. Since this is a simulation of the actual product, we do not give the specific parameters of the coil and cold shield. We take the center axis of the magnet as the z-axis and the center point of the magnet as the origin of the coordinate to establish the Cartesian coordinate system and the cylindrical coordinate system. The coordinates in the Cartesian coordinate system are represented by (x, y, z), and the coordinates in the cylindrical coordinate system are represented by Through analysis, it is known that the force on the cold shield occurs at a certain time in the early stage of loss of superconductivity, because the magnetic field generated by the coil is the largest at this time. After the electromagnetic force reaches the peak at a certain time, it gradually decays with the decay of the current. Figure 3 is the simulation of the eddy current density distribution on the cold shield at t = 1.2 s. Figure 4 is the curve of the axial distribution of the Lorentz force density f on the cold shield at t = 1.2 s. Because f is a vector, the graph contains two curves. fz is the Lorentz force density in the z direction, and fρ is the Lorentz force density in the ρ direction. By analyzing the changes of the Lorentz force density at different times and different positions, we can determine whether the force on the cold shield is within the bearing range, thereby providing guidance for structural improvement.

Claims

1. A finite element analysis method for the quenching process of a superconducting magnet, characterized in that, Includes the following steps: (1) The current signal is obtained by sampling the current signal through the magnet monitoring equipment, or the current-time relationship curve I(t) in the magnet coil after magnet quench is obtained by the ellipsoidal diffusion model. (2) Using I(t) as the excitation source, the coordinates at the start of the quench are set as t = 0. The initial temperatures of each part of the magnet before the quench are determined. The eddy current density generated on the conductor in the magnet at time t = t1 is calculated using a time-domain numerical simulation algorithm. The eddy current density at each time is numerically simulated using the finite element method, and the following electromagnetic field control equations are used for finite element modeling: In the conductor region: In the non-conducting region: In the above formula, A is the vector magnetic potential generated by the eddy current density, V is the scalar potential, and v is the reciprocal of the permeability. Coordinates The electrical conductivity of the metal, J s The current density inside the coil, symbol Gradient calculation is represented; a tetrahedral mesh is used for meshing, and edge basis functions are used as interpolation functions for the magnetic vector potential; the eddy current density in the conductor is obtained by the following formula: (3) Calculate the coordinate point of any point on the conductor in the magnet system using the following formula. Heat generated by eddy current density In the above formula, Coordinates conductivity at that point for The modulus of the eddy current density at that location; (4) Solve the following heat conduction control equation to obtain the temperature distribution inside the conductor at time t = t1: In the above formula, T is temperature, ρ is material density, c is specific heat of the material, x, y, z are coordinates in three directions in the Cartesian coordinate system, and k is... x ,k y ,k z Coordinates The thermal conductivity in the x, y, and z directions is used; in the heat conduction control equation, a tetrahedral mesh is used for subdivision, and nodal basis functions are used as interpolation functions for the temperature field; (5) Based on the temperature distribution obtained in step (4), redetermine the conductivity of the conductor at each coordinate point, and then repeat steps (2)-(4) to calculate the eddy current density and temperature at the next moment until the calculation ends; when the maximum eddy current density in the conductor satisfies When the simulation ends, ε is the set threshold. any coordinate point on the conductor within the magnet system Lorentz force density at the location Calculated using the following formula: In the above formula, coordinate point Eddy density at that location coordinate point The magnetic field at that location.

Images