# Presentation: 848880

## Modeling The Quench Process in Super-conducting Coils

- _J.-M. Gery_, C. Emson, J. Buan, A. Michaelides, C.P. Riley

***CHATS-06, September 5-7 in Berkeley, California***

## Introduction

- This presentation shows the use of the transient thermal analysis, QUENCH, to determine the temperature distribution in a super-conducting coil during a “quench”.
- Quenching can occur:
- if the temperature of the superconductor becomes too high
- it experiences a high magnetic field
- the current density exceeds its critical value
- usually a combination of all three.
- Since super-conducting materials are poor conductors when not in SC regime, the wire in a SC coil is generally composed of super-conducting filaments embedded in a metallic conductor.
- This allows a low electrical resistance path for current that was being carried by the superconductor prior to the quench.

## Introduction

- The model is created using the OPERA-3d Modeler.
- The temperature rise and quench is initiated by a film heater.
- Resistive heating will start generating additional volume heat sources when the nonlinear critical temperature is reached.
- The analysis includes evaluation of the magnetic field produced by two solenoid conductors which are also connected to an external protection circuit.
- In the next release of OPERA-3d, coupling of the temperature distribution with transient electromagnetic fields will be simulated by a QUENCH analysis coupled with an ELEKTRA/TR analysis.

## The Quench process in Super-conducting Coils

- OPERA-QUENCH uses a finite element method to simulate the transient thermal behavior of super-conducting magnets. Coupled transient electromagnetic - thermal - circuit simulation is used.
- Functional material properties are paramount in the modeling of the quench process in a super-conducting magnet coil.

## The Quench process in Super-conducting Coils

- Introduce Quench process using a film heater on the inner radius.
- Once initiated, the quench process will propagate, as the quenched volume becomes resistive and produces ohmic heating.

## The Quench process in Super-conducting Coils

- Mass density and specific heat capacity of the bulk material representing the coil depend on the properties of super-conducting material NbTi and copper.
- The bulk thermal conductivity is nonlinear and highly anisotropic.
- The critical current density of super-conducting material is a function of the temperature and the magnetic field.

## Building a Model in OPERA

- To build a model using OPERA, the following steps are performed:
- Build conductors – need to be meshed to model temperature rise
- Define material properties – function of T and field B.
- Define Critical Current – function of T and field B
- Define protection circuit
- Define a heat source that initiates the quench process

**Notes:**
In this example, the coil is split into two sections in the radial direction, to allow different mesh size in the inner and outer sections. The first section ranges from radius 6 cm to 6.84 cm, and is named coil1. The second one ranges from radius 6.84 cm to 7.68 cm, and is named coil2. Both sections have the same height, 6 cm, in the Y direction, in a range from - 3 cm to 3 cm. A finer mesh is applied to the first section, since the quench process starts in this volume.

## Build Conductors

- Build the coil
- Simple parameterized

- Add the heater element
- Short pulse of heat toinitiate the quench

**Notes:**
As QUENCH is a thermal solver, only solid objects - in this case the solenoid
coil volume - are included in the geometry. The surrounding environment
is modelled by a thermal boundary condition.

Mesh: The quench process will mostly propagate in the azimuthal direction at the
early stage, because the thermal conductivity in the direction is much
greater than those in transverse directions. Initially, temperature rise only
occurs over very small distances in the axial and radial directions. Therefore,
an anisotropic mesh with very fine element sizes in the transverse
directions has to be used to model the extremely steep temperature gradient.
If the mesh is not sufficiently fine, numerical noise will be excessive
resulting in non-physical temperature distributions, although this does not
affect the solutions later in the transient.

Film Heater The film heater is modelled by a Heat Flux boundary condition. Therefore
the contacting surface area between the heater and the superconductor coil
has to be created and labelled. Near the film heater, a smaller mesh size in the radial direction is needed since heat generated through the film heater can only penetrate very slowly
radially while propagating in the azimuthal direction quickly.

## Coil specification

- Automatic creation of the winding data from OPERA’s standard conductor set
- Solenoids
- Racetracks
- Bedsteads
- Constant perimeter end
- Arcs and bars
- curvy bricks

## Conductors with Symmetry

- Only quarter of model needs to be analyzed
- 	The quench process will mostly                          propagate in the azimuthal direction                          at the early stage, because the                                     thermal conductivity in the direction                          is much greater than those in                              transverse directions. Initially,                         temperature rise only occurs                                         over very small distances in the                                     axial and radial directions.                                    Therefore, an anisotropic mesh                              with very fine element sizes in the                                transverse directions has to be used                          to model the very steep temperature                         gradient. If the mesh is not sufficiently                         fine, numerical noise will be excessive                  resulting in non-physical temperature distributions.

**Notes:**
Although QUENCH is a thermal solver, the magnetic field distribution has
to be considered in the model, because it also affects the critical current
density. As well as becoming meshed volumes for the thermal analysis, the
solenoid conductors included in the model are used to compute the field
using the Biot-Savart expressions. The conductors are linked to windings
in an external circuit for modelling the transient currents through the conductors.

## Apply material properties to coils

- Set anisotropic thermal conductivity

- Set heat capacity and density

- Set wire properties, including critical current

**Notes:**
Thermal conductivity
For the azimuthal thermal conductivity of the bulk material, the conductivity of copper, Cu_Kappa(T), will be scaled by the copper factor. It is assumed that this is significantly higher than the conductivity of NbTi and
dominates the thermal conduction in this direction.
The bulk properties radially and axially are assumed to be directly from the tables of measured values. (Note that the data used in this example are fictional, but with similar characteristics).

Mass density The mass density of the bulk material is computed from those of copper and
superconductor in the appropriate ratio. Both of them have been defined to be constants in this example, but could be extended to nonlinear properties if the density varied significantly with temperature.

Specific heat capacity
The nonlinear heat capacity of the bulk material is specified.

Electrical conductivity
The nonlinear electrical conductivity of the wire when operating in the “normal” regime will be specified.

## Nonlinear Material Properties

- Functional material properties allow the user to describe the variation of properties such as electrical or thermal conductivity with quantities such as magnetic field or temperature.
- Thermal and electrical conductivity are function of Temperature
- Critical current function of Temperature and Field

## Variation of Electrical Conductivity with T(K) 

**Notes:**
Here is a plot of the tabulated function defiend by NbTi_Jc.table.

X:   T   arrange from 0 to 10 degree
Y:   B  set 0

## Variation of Thermal Conductivity with T(K)

## Jc as a function of B(T) and T(K)

**Notes:**

X: T arrange from 0 to 10 degree
Y: B arrange from 0 to 5 Tesla

## Nonlinear Material Properties

- Functional
-    eg.    10a+b*log(t)+c*log(t)^2+d*log(t)^3....(Typical NIST format for material properties)
- Conductivity as function of current density...
- Tabulated
- User defined tabulated function eg. JC(T;B)

## Example Function Table format

-  11           9           1           1
-  1 T [1.0]
-  2 BFIELD [FLUXU]
-  3 NBTI_JC [CURDU]
-  0
-  0 0.0E4  4.75E+06
-  1 0.0E4  4.66E+06
-  2 0.0E4  4.39E+06
-  3 0.0E4  3.93E+06
-  4 0.0E4  3.32E+06
-  5 0.0E4  2.68E+06
-  6 0.0E4  2.04E+06

## Nonlinear tensor and vector hysteresis

- Full nonlinear tensor representation of any material property
- Eg. a term in the magnetic permeability tensor could be specified as

## Apply material properties to coils

- Set nonlinear material properties as Tables

**Notes:**
Critical current The critical current is used to decide whether material is under superconducting or normal state. It will be specified by the two-argument tabulated function of critical current density in NbTi, the area of wire cross section and the user-defined variable of superconductor factor where T and B is the temperature and the flux density of the magnetic field at evaluation points. The magnetic field is generated by the currents through the solenoid conductors.

## Protection Circuit

- Rprot1 = 1 Ohm
- Rprot2 = 1 Ohm
- L11 = 0.0443H
- L22 = 0.05356 H
- L12 = L21 = 0.04206H

**Notes:**
An external circuit is included to model the transient current process. 
As mentioned previously, the full coupled transient electromagnetic model for this example is not available through the Elektra/TR solver yet, although the magnetic field from the solenoid conductors can be evaluated by QUENCH solver itself. Therefore the stored energy in magnetic field and magnetic coupling between the sections of the coil must be modelled by self and mutual inductances provided by the user. Also, initial currents of the windings (prior to the quench) should be set.
The resistance of the two sections of the coil will each be linked to the resistance of the respective wire material. Therefore the winding name and the base name of the respective material must match each other. The matching has be done when a section of the coil and its respective winding were created by an advanced SOLENOID command. As only a quarter of the coil is modelled, the parameter SYMMETRY of the windings has to be set 4 to scale the wire material resistance to the resistance of the whole winding. 
More circuit elements, used as protection resistors, are also included in the quench circuit.
In this example, the quench circuit and circuit parameters are displayed as shown.

## Defining Circuit Components

- Winding properties are set, including number of turns, symmetry, and initial current.

## Defining Circuit Components

- Circuit loops are then defined as collection of individual components

## Joule heating +

- Additional heat sources may be introduced
- Rate dependent losses

## Running the Analysis

- A transient thermal model is run.
- At each time step:
- Solve for temperature
- Update current equations using temperature dependant properties
- Update heat generation due to currents
- If non-linear option selected:
- Temperature solution includes temperature dependant thermal properties

## Propagation of the quench process

- 100 ms

- 200 ms

- Along the azimuth, the copper conductivity is dominant

- 500 ms

## Slide 26

## Slide 27

## Slide 28

## Benchmark Model

- PT55 Polarized target magnet
- Built and measured at Rutherford Appleton Lab, UK

## PT55 magnet parameters

- 60 degree access and exit angles
- Main coil pair – 8168 turns/coil
- Correction coil pair – 1365 turns/coil
- C361 super-conducting wire (2:1 Cu:NbTi)
- Peak Field 5.6T
- Total inductance 98Henries
- Central field 2.5T

## Measured and Calculated

## Results

- User selectable tabulated results
- Current, resistance, temperature, voltage
- OPERA-3d database
- Full solution saved at specified times
- For subsequent calculation and display using the OPERA-3d Post processor
- Data available includes: Temperature, Joule heat density, B, dB/dt, thermal conductivity ....

## Temperature at a particular time

## Resistive loss in the normal zones

- Note that all the coils quench in this example, and that the coupled coils quench because dB/dt losses raise them above critical temperature.

## Summary

- Evolving requirements for material characterization
- Functional magnetic material properties
- Non-linear tensor and vector hysteresis properties
- Applications
- Quench of super-conducting magnet coils
- Heating effects in damper bars
- Induction Hardening of automotive parts
- Modeling the magnetization of hard materials and their degradation ‘in service’

## Summary

- First release of QUENCH (now available)
- Transient thermal
- Circuit equations
- The inductance matrix must be provided
- Second release (due later in 2006)
- Transient thermal
- Circuit equations
- Coupled to transient EM simulation (Elektra)