Design Principles of Superconducting Magnets

Transcription

1 1 Design Principles of Superconducting Magnets Aki Korpela Tampere University of Technology

2 DESIGN PRINCIPLES OF SUPERCONDUCTING MAGNETS 2 Content of the presentation Background Short-sample measurement of a superconducting wire Power law Critical current of a superconducting magnet Design case of low temperature superconductor (NbTi) solenoidal coil Design case of high temperature superconductor (Bi-2223/Ag) solenoidal coil Optimization of coil geometry Design case of low temperature superconductor SMES

3 3 Background Critical quantities of superconductors are: current density J, magnetic flux density B, temperature T. These three parameters constitute a critical surface, inside of which a material is in the superconducting state. Piercing of this surface is called as a resistive transition. The maximum values of J, B and T that maintain superconductivity at certain operation conditions are called as: critical current density J c, critical magnetic flux density B c, critical temperature T c.

4 Short-sample measurement of a superconducting wire 4 In a typical short-sample measurement of a superconducting wire the critical current I c is measured at constant temperature T and constant magnetic flux density B. As a result we get I c (B)-dependence of the wire at a certain temperature, which is fundamental data concerning the superconducting magnet design. Short-sample data at 4.2 K for NbTi wire OK42 manufactured by Luvata.

5 Short-sample measurement of a superconducting wire 5 Critical current is determined with an electric field criterion E c. In LTS measurements the criterion is typically E c = 0.1 V/cm. In HTS measurements the criterion is typically E c = 1 V/cm. Adjacent measurement is for Bi-2223/Ag tape (HTS material) at 77 K and self-field. In this case the distance between voltage taps is 2 cm, so I c ( 120 A) is determined with the voltage criterion of 2 V.

6 Short-sample measurement of a superconducting wire: power law 6 Voltage (V) - current (I) measurement of a superconducting wire can be modelled with so called power law, where critical current I c is determined from the measured V(I)-curve with voltage criterion V c, and exponent n describes steepness of the resistive transition: I V Vc I c n. Adjacent figure shows power law fitting for measured data: V c = 2 V, I c = A, n = 8 (+), n = 10 (o), n = 12 ( ).

7 Summary: short-sample measurement of a superconducting wire 7 The idea is to find out I c and n at certain temperature and magnetic flux density. When I c and n are known, V(I)-characteristics of a sample can be modelled with power law. Cool down the sample to a certain temperature. at first the magnetic flux density applied on the sample is zero, B = 0 while B B max measure the voltage - current curve of the sample determine I c from measured V(I)-curve with electric field criterion E c determine exponent n by fitting power law to measured V(I)-curve B = B + B end Change the sample temperature and repeat the while loop. As a result we get I c (B,T) and n(b,t) data of the superconducting sample, after which the V(I)-behaviour of the sample can be modelled with the power law.

8 8 Critical current of a superconducting magnet Critical current of a superconducting magnet, I cm, is the maximum current that enables a stable operation at certain temperature. The computational determination of I cm for a certain coil geometry requires that: short-sample data of the wire is available. magnetic flux density distribution in the coil has to be computed. Basically the determination of I cm requires the solution of heat conduction equation inside a superconducting coil: T T T Q T, Iop Cp T, t where and C p are the thermal conductivity and the volumetric heat capacity of the coil, and Q is the heat generation inside the coil.

9 Basic differences between LTS and HTS magnet design 9 Resistive transition in LTS is much steeper than in HTS. Typical value for exponent n is about 100 for LTS and about 10 for HTS. In practice a steep transition means explosive heat generation with overcritical currents. I c (B)-dependence of LTS wire is isotropic, but in HTS wires the situation is anisotropic. In isotropic situation I c depends only on the magnitude of B. In anisotropic situation I c depends on the magnitude and direction of B.

10 10 Critical current of LTS magnet (1/3) The determination of I cm for LTS magnet simplifies to: determine the critical current of the coil turn that is exposed to the highest magnetic flux density. We don't have to care about the solution of heat conduction equation inside the coil, because the resistive transition in LTS is so steep that any cooling cannot tolerate the explosive heat generation at overcritical currents. We don't have to care about the direction of magnetic flux density, because the I c (B)- dependence of LTS wires is isotropic. So, in order to determine I cm for LTS magnets, we have to compute the magnitude of B applied to each turn of the coil at certain operation current I op.

11 11 Critical current of LTS magnet (2/3) In solenoidal coil the maximum of magnetic flux density, B max, is located at the axial mid-plane of the coil's inner radius (red-coloured turns in the figure). If there is not any magnetic material in the vicinity of the coil, the dependence between B max and I op is linear. I op (B max ) is called a load line.

12 12 Critical current of LTS magnet (3/3) The critical current of LTS magnet is found from the intersection of load line and shortsample based I c (B)-dependence of the wire in the coil. Typically 80-90% of the short-sample I c is achived, when the wire is wound as a solenoidal coil. In adjacent figure the intersection of load line and I c (B)-dependence of the wire in the coil is at the point (4.8 T, A). In other words: the operation current of A is the critical current of the coil. when the operation current of the coil is A, the magnetic flux density in the mid-plane of the inner radius is 4.8 T.

13 13 Critical current of HTS magnet (1/6) Preceding method to determine the critical current of LTS magnet cannot be applied to HTS magnet, because: HTS materials have slanted resistive transition due to which HTS magnets can be momentarily operated with overcritical currents. HTS materials have anisotropic I c (B)-dependence due to which the critical current of the magnet cannot be determined by investigating a single turn of the coil. In LTS magnets, the loss of superconductivity typically occurs, when critical current is exceeded in the coil turn exposed to the highest magnetic flux density. In HTS magnets, the loss of superconductivity is viewed as a global temperature increase inside the coil rather than in a single turn. In the following we concentrate on determining the critical current of a solenoidal Bi-2223/Ag coil.

14 14 Critical current of HTS magnet (2/6) Because I c of HTS wire depends on the direction of B, the short-sample data of HTS wire is more complicated than the one of LTS. Adjacent figure presents I c (B)- and n(b)-dependence at 20 K for Bi-2223/Ag wire manufactured by American Superconductor. Curves from top to bottom represent the orientation of B from parallel (B II ) with an interval of 10 o to perpendicular (B ).

15 15 Critical current of HTS magnet (3/6) In solenoidal coil the maximum of magnetic flux density, B max, is located in the axial midplane of the coil's inner radius. However, due to anisotropic I c (B)-dependence, the point of B max is not the weak spot in HTS coils. Typically the lowest I c in HTS coils is found from the turn that is exposed to the highest perpendicular magnetic flux density B (in solenoidal coil near the coil flanges).

16 16 Critical current of HTS magnet (4/6) When the critical current of each coil turn is known, the electric field E in each coil turn can be computed from the power law: n Iop E Ec. I c When electric field E and current density J of each coil turn are known, the power loss density Q of each coil turn can be computed: Q = EJ.

17 17 Critical current of HTS magnet (5/6) In order to accurately determine the critical current of HTS coil, the heat conduction equation inside the coil has to be solved: T T T Q T, Iop Cp T. t The solution cannot be found analytically, but numerical approach is required. Finite Element Method (FEM) is one potential method to solve partial differential equations numerically. The idea is that the coil is operated with constant I op, and then the temperature distribution inside the coil is computed by solving the heat conduction equation as a function of time. If the minimum temperature inside the coil starts to increase I op > I cm. If the maximum temperature inside the coil starts to decrease I op I cm. Thus, the magnet critical current, I cm, has to be searched iteratively in case of HTS.

18 18 Critical current of HTS magnet (6/6) The accurate determination of HTS magnet critical current requires the solution of heat conduction equation inside the coil. This is a quite complicated computational task. However, the critical current of HTS magnet can be estimated by monitoring the average electric field, E ave, between the coil terminals: E ave = V/l, where V is the voltage between the coil terminals, and l is the length of the wire in the coil. These operation temperature dependent critical current estimates for Bi-2223/Ag coil are: 20 K: I op I cm, when E ave = 0.1 V/cm. 77 K: I op I cm, when E ave = 0.3 V/cm. 4.2 K: I op I cm, when E ave = 0.01 V/cm. However, it has to be emphasized that these critical current criteria are only estimates and apply only for properly cooled Bi-2223/Ag coils.

19 19 Summary: critical current of a superconducting magnet Generally speaking, the accurate determination of critical current of a superconducting magnet requires the solution of heat conduction equation inside the coil. However, this is unnecessarily complicated way to solve I cm in case of LTS, because: LTS wires have very steep resistive transition and isotropic I c (B)-dependence. I cm of LTS coils can be determined as I c of the coil turn exposed to B max. The accurate determination of I cm in case of HTS requires the solution of heat conduction equation, because: HTS wires have slanted resistive transition and anisotropic I c (B)-dependence. I cm of HTS coil can be estimated with some voltage-based criteria, but these are always temperature and material dependent.

20 20 Optimization of coil geometry (1/4) So far we have only been dealing with the determination of magnet critical current I cm, which is a fundamental parameter related to superconducting magnet design. Other, usually application-dependent, design parameters are: coil volume (in order to save superconducting wire and hence money) stored energy (e.g. SMES), geometry requirements (e.g. MRI), magnetic flux density distribution requirements (e.g. MRI, particle acceleration), losses in AC use (e.g. SMES), etc... Generally speaking, the design of a superconducting magnet is a nonlinear optimization problem.

21 21 Optimization of coil geometry (2/4) The aim of superconducting magnet design is to find an optimal coil geometry for a given application. For example in solenoidal coil the task is to find optimal values for inner radius a, outer radius b and axial length d. The meaning of "optimal values" depends on the application. If we for example want to minimize the coil volume and have the magnetic flux density of 5 T in the center of the coil bore, then we try to find such a, b and d that produce the required magnetic flux density with the smallest possible coil volume.

22 22 Optimization of coil geometry (3/4) The design of a superconducting magnet can be formulated as a constrained nonlinear optimization problem. Let's consider the following task to design a solenoidal coil for superconducting magnetic energy storage (SMES) application: the coil should be as small as possible. the stored energy, W, should be at least 200 kj when operated with I cm. the maximum magnetic flux density, B max, in the coil is 5 T when operated with I cm. Coil design can be stated as a constrained optimization problem: minimize V(a,b,d) subject to g(a,b,d,i cm ) 0, where V is the coil volume, and g is the vector of the constraint functions.

23 23 Optimization of coil geometry (4/4) Thus, the design task is to minimize V(a,b,d) such that: W(a,b,d,I cm ) J (constraint function g 1 ), B max (a,b,d,i cm ) 5 T (constraint function g 2 ). In other words the vector of constraint functions can be written as: g 5 g1 abdi,,, cm 2 10 W abd,,, Icm abd,,, Icm 0. g abd,,, I B abdi,,, 5 There are several algorithms to solve such optimization problems. One of the most effective ones is sequential quadratic programming (SQP) that is an iterative optimization algorithm for constrained nonlinear problems. 2 cm max cm "Iterative" means that an initial value for the coil geometry has to be set in the beginning of iteration, after which the algorithm tries to find a feasible solution that minimizes the coil volume.

24 Optimization of coil geometry: case of 200 MJ Nb 3 Sn SMES coil 24 The task was to minimize the volume of a solenoidal coil such that: the strored energy should be at least 200 kj when operated with I cm. the maximum magnetic flux density, B max, in the coil is 5 T when operated with I cm. the maximum tolerable AC losses during charging the coil (30 s) is 1.5 W. the maximum stress values are 100 MPa for hoop stress and 50 MPa for radial stress. Adjacent figures show the development of the solution during iteration with SQP.

25 25 Summary: optimization of coil geometry Usually superconducting magnet design means that the coil geometry has to optimized for a certain application. Regardless of the application, the design process can be formulated as a nonlinear optimization problem, which can then be solved with some optimization algorithm. In order to build computer aided tools for the design of superconducting magnets, a software for solving partial differential equations by means of finite element method is practically required. For my own PhD research the partial differential equations of field computations and thermal analysis were solved with Comsol Multiphysics. The optimization problem was solved with SQP, which was a built-in package in the Optimization toolbox of Matlab. Generally speaking, the design of a superconducting magnet is a nonlinear optimization task with some particular constraints set by the special features of superconductivity.

26 26 Applications utilizing superconducting magnets Design parameters of superconducting magnets are application-dependent. However, the basic principles related to short-sample data and critical current of the magnet are valid regardless of the application: Superconducting Magnetic Energy Storage (SMES), Magnetic Resonance Imaging (MRI), Induction Heating, Generators/Motors (for example for 10 MW wind power plants), High Energy Physics.

27 27 Thank you for your attention.