# Presentation: 906540

## Simulation of Electromagnetic and Thermal processes in Rutherford-type Cables

- E.Barzi, _V.Lombardo_,
- J.McDonald, D.Turrioni, A.Zlobin.

- Chats ‘06

- 08/02/05

## Introduction

- A Fast-Multipole-Method (FMM) has been implemented in a network model for analyzing the transient electromagnetic response of Rutherford-type cables (J. McDonald et al., ASC 2004).
- Thanks to the reduction by an order of magnitude of the data storage and operation count required, this code made feasible modeling long cables containing many strands.
- Since then, the FMM-network model has been expanded to include thermal analysis, and graphical tools to improve software usability.
- The code is currently being commissioned to simulate quench dynamics of a localized normal zone in a 28-strand cable.

## Outline

- Quick overview of the network model and Fast Multipole Method
- Summary of software features:
- Electromagnetic analysis
- Thermal analysis
- Inputs/Outputs
- Simulations:
- Ramped transport current in a short cable with soldered ends
- Quench Dynamics: quench or recovery after thermal disturbances
- Conclusions, ongoing and future works

## Network Model of Rutherford Cable(A. Verweij, PhD Thesis)

-  Cable of length L, containing Ns (even) number of strands
-  Cable has a pitch length Lp
-  The cable can be divided into Nb bands of length
-  Each band can be decomposed into a collection of subnetworks.

## Subnetworks structure

- Within each subnetwork, the electrical contacts between strands are represented by ohmic resistors
- Each section of strand is represented by a piece of wire with a resistance and a self inductance
- Each strand section is linked to every other strand section in the cable by a mutual inductance

- The equations determining the unknown N*=Nb(5Ns-3) currents are derived by applying Kirchoff’s laws to loop and nodes within the network.

## Deriving the equations

- The problem can be formulated in terms of transmission-line equations and can be solved directly for the currents in the strand sections
- Is = vector of currents in the strand sections
- Us = vector of voltages at contact nodes
- deltaIs = vector of current changes along the strand sections (due to inter-strand current transfer)
- deltaUs = vector of voltage changes along the strand sections
- M = inductance matrix
- Rs = diagonal matrix containing the resistances of the strand sections
- G = conductance matrix describing the electrical contacts between strands
- ε ext = vector of externally-induced electro-motive forces along the strand sections

## Solving the equations(Dr. Jason McDonald)

- For ‘classical’ iterative methods (such as conjugate-gradient) matrix-vector multiplications become a bottleneck in computation, requiring N^2 operations, with N=NpNs(2Ns-2).
- The easiest way to solve the problem is to completely neglect all the strand sections beyond a certain distance away; but this is impractical because of the long-range nature of the inductive interaction: one would need to take a huge number of neighbors to obtain good accuracy
- A better way is to directly evaluate all the interactions within a small local region, and then approximate the influence of all distant sources by a short series; this is called the Fast Multipole Method or FMM (Rokhlin, Greengard). This allows to reduce the operation count and storage requirements to N.
- The FMM allows the inductive-voltage effects to be included accurately (error << 1%) without the need to store, access, or invert high order non-sparse matrices.

## Toward a more usable code

- In the original Fortran version of the code, any parameter change had to be made inside the code as all parameters values were built in the code itself.
- Need of a Fortran compiler on the machine
- Need of external software for offline data analysis
- Software Outputs:
- Adjacent Currents
- Cross-over Currents
- Strand Currents
- Strand Temperatures

## Graphical User Interface

- A Matlab GUI has been developed to improve usability of the software in terms of:
-  Parameters input
-  Code operation
-  Data Analysis

## Input Parameters

- ELECTROMAGNETIC
- 		AdjacentResistance
- 		CrossOverResistance
- 		RampRate_Field_X
- 		RampRate_Field_Y
- 		RampRate_Field_Z
- 		InitialField_X
- 		InitialField_Y
- 		InitialField_Z
- 		RampRate_Current
- 		InitialStrandCurrent
- NUMERICAL/SIMULATION
- 		N.of SimulationSteps
- 		TimeStep
- 		Tolerance

-  THERMAL
- HeatTransferCoeff
- AdjacentConductancePerLength
- CrossOverConductancePerLength
- BathTemp
- DisturbanceEnergy
- DisturbanceLength
- DisturbanceDuration
- DisturbanceStrandNumber
-  CABLE GEOMETRY
- StrandDiameter
- CableLength
- PitchLength
- NumberOfStrands

## Ramped transport current in a short cable with soldered ends

- Soldered
- regions

## Current imbalance (1)

**Current imbalance (1)**

## Current imbalance (2)

- Whereas running the code with the same values of Ra and Rc either in soldered and middle regions produces a flat current distribution, the simulation predicts some current imbalances when there is a discontinuity in contact resistances.

## Distribution of strand currents along the cable length

**Distribution of strand currents along the cable length**

## Thermal Analysis

- As in the electromagnetic case, the thermal analysis can be formulated using a network model in which the strand sections and the contacts between them are represented by thermal conductances.
- The heat-balance equation is solved at each time step for the temperatures of the strand sections.
- In the coupled analysis, the electromagnetic and thermal modules exchange data with one another so that material properties and heat sources can be updated at each time step.
- The code allows to simulate a heat-up process in one of the strands depositing a certain amount of energy for a certain amount of time in the strands itself.

## Quench or recovery

- 	In the following simulations a localized disturbance has been applied to a strand and the thermal transient has been followed along the cable length.
- 	Cases of quench and recovery are discussed.
- _Simulation parameters:_
- Disturbance energy range inspected: from 1e-6 to 10e-3 J
- Disturbance duration: 5e-4 s
- Disturbance length: 20e-3m
- Magnetic Field: 12T
- Bath Temperature: 4.2K
- Thermal inter-strands Conductances: 5e-3 W/m
- Strand Diameter: 1mm
- Number of Strands: 28
- Time step: 1e-4s
- Heated up strand: 5

## Recovery (1) – Strand Temperatures

- _Strand n.5 (heated up)_

- _Strand n.4 and 6_

**t = 5e-4s**

- _Strand Current=500A_
- _Disturbance Energy=0.77mJ_

## Recovery (2) – Strand currents

- _Strand n.5 (heated up)_

- _Strand n.4 and 6_

**t = 5e-4s**

- Current in strand n.5 starts to drop as the
- strands temperature goes up.

- Current diffusion in adjiacent strands

## Recovery (3)

**t = 7e-4s**

**t = 9e-4s**

## Recovery (4)

**t = 15e-4s**

**t = 20e-4s**

## Recovery (5)

**t = 30e-4s**

**t = 40e-4s**

## Recovery (6)

- After the thermal disturbance, the cable recovers completely its superconductive status.
- Temperatures slowly gets back to ~4.2K and strand currents stabilize around ~500A after a
- small transient.

**t = 60e-4s**

**t = 100e-4s**

## Quench (1)

**t = 2e-4s**

- Strand n.5 (heated up)

- Strands 4 and 6 starts to warm up due to thermal diffusion

- _Strand Current=500A_
- _Disturbance Energy=0.78mJ_

## Quench (2)

- Strand n.5 is warming up to ~13K, way over its critical temperature

**t = 10e-4s**

## Quench (3)

- Strands 4 and 6 warming up over their critical temperatures due to thermal diffusion from strand 5

**t = 30e-4s**

## Quench (4)

- Strands 3 and 7 warming up over their critical temperatures due to thermal diffusion from strands 4,5 and 6

**t = 40e-4s**

## Quench (5)

**t = 50e-4s**

**t = 100e-4s**

- All the strands of the cable  warms up
- over their critical temperatures showing
- a cable quench.

- The plot shows how the _quench propagates_
- _along the whole cable length_.

## MQE evaluation example

## Conclusions, ongoing and future works

- Given the software flexibility in terms of
- Cable geometry specification
- Electromagnetic/Thermal parameters
- Given the reasonable calculation times achieved
- 	(even for high numerical accuracy ~10^-8)
- simulation possibilities are endless.
- Among ongoing and future works:
- Simulations on longer cables.
- MQE evaluation for different cable geometries and disturbance amplitude.
- Apply a disturbance energy and see if it is enough to have a cable quench, if not increase the disturbance energy and repeat the simulation.
- Effect of perturbation time (from 10^-6s to 0.1s)
- Effect of magnetic field.
- Non-constant resistances along the cable. (simulation of cut or broken strands)
- Feedback from experimental data are needed.