3 Sn strands and magnets Author Carmelo Barbagallo Supervisors Matthias Mentink and Arjan Verweij TEMPEPE 30 th August 2019 ShortTerm Internship Programme MarchSeptember 2019 ID: 934190
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Slide1
Inter-filament coupling currents modelling in superconducting filamentary Nb3Sn strands and magnets
Author: Carmelo BarbagalloSupervisors: Matthias Mentink and Arjan Verweij TE-MPE-PE, 30th August 2019Short-Term Internship Programme – March-September 2019
1
Acknowledgments
: Lorenzo Bortot, TE-MPE-PE
Slide2Introduction on Inter-Filament Coupling Currents (IFCCs)
Aim of the study and roadmapProposed analyses:Single round strand modelling:Study 1: Equivalent Magnetization Formulation;
Study 2: Equivalent Resistivity Formulation – 1
st approach;
Study 3: Equivalent Resistivity Formulation –
2
nd approach.Stack of Rutherford cables:Homogenized vs Multi-strand model;Modified IFCCs equation for the homogenized model.Conclusions (1/2) – Single round strand and Rutherford cable modelling
Outline 1/2 – Single strand and Cable stack
2
Slide3Introduction on magneto-thermal analysis of FRESCA2 magnet;
Aim of the study and roadmap – FRESCA2 magnet;Proposed analyses (Energy Extraction and CLIQ simulations):
Homogenized SIGMA model;
Homogenized SIGMA restructured model;
Multi-strand model;
Conclusions (2/2) –
FRESCA2 magnet;Overall conclusions.Outline 2/2 - Magnet3
Slide4Introduction on IFCCs
4[1] E. Ravaioli, B. Auchmann, G. Chlachidze, M. Maciejewski, G. Sabbi, S. E. Stoynev, and A. Verweij, Modelling
of Interfilament Coupling Currents and Their Effect on Magnet Quench Protection,
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 27, NO. 4, JUNE 2017.
Superconducting wires
are composed by a large number of filaments embedded in a normal-conducting
matrix (usually copper).When they are exposed to a time-varying external and transverse magnetic field , a magnetic field
is induced, which opposes the applied magnetic field change. The resulting total magnetic field
is [1]:
where
[m]
is the filament twist
pitch,
[
Ω
m] is
the effective transverse resistivity of the
strand matrix
and
[TmA-1] the magnetic permeability of vacuum.
CERN
The induced magnetic field
is generated by local coupling currents between superconducting filaments, i.e., Inter-Filaments Coupling Currents (IFCCs), which flow through the resistive wire matrix generating ohmic losses.
The Inter-Filaments Coupling Currents (IFCCs) develop with a characteristic time constant:
Slide5Introduction on IFCCs
5[2] M. Wilson, Superconducting magnets. Oxford University Press, 1983.[3] B. Auchmann, L. Bortot, I. C. Garcia, L. D’Angelo, H. De Gersem, M. Mentink, S. Schöps and A. Verweij,
Ad-hoc Homogenisation for Interfilament Coupling Currents, STEAM collaboration - BMBF project, Internal report, 2019.
By means of the internal field
, we can calculate the
magnetization
(in Am
-1
) due to the
IFCCs
from direct integration of the coupling current distribution
[2]:
The magnetization is added as an additional source term in the Magnetoquasistatic (MQS) partial differential equation (PDE), i.e.,
This magnetization term is due to small closed current-carrying loops, and the corresponding magnetizing current
is
divergence-free
[3].
where
is the magnetic vector potential,
the conductivity and
the current density applied to the strand.
where
is the reluctivity and
the IFCC decay time constant.
Assuming no further eddy-current effects and considering the relation
the
following MQS equation is obtained and solved for individual strands by using Finite Element Method (FEM):
Slide6Aim of the study and roadmap
6Aim of the study: Understanding of different magnetization formulations;Modelling of IFCCs physics in Rutherford cables and magnets;
Show the impact of the homogenization on IFCCs behavior.
MQS equation
FEM simulation
Rutherford cable
FRESCA2 magnet
Single round strand
Cable stacks
Homogenized model
Multi-strand model
Homogenized model
Multi-strand model
Slide7Overview on different formulations to model IFCCs
7Equivalent Magnetization
approach:
M. Wilson, A. Verweij
(W-V)
formulation
[2]For single strand, multi-strand cable stack, and multi-strand magnet models.M. Wilson, A. Verweij formulation for homogenized models (W-V-homo) For homogenized cable stack and homogenized magnet models.3. Modified (W-V-homo) formulation:
For
homogenized cable stack and homogenized magnet models.
Equivalent Resistivity
approach:
1
st
approach (M. Mentink)
(M-M)
:
2
nd
approach (C. Barbagallo, L, Bortot)
(B-B)
:
Both for single strand, multi-strand cable stack and multi-strand magnet models.
(W-V-homo-mod)
Slide8Single round strand modelling8
Slide9Single round strand modelling
9
Strands submitted to an external varying magnetic field (ramping from 0 T at 1 T/s), obtained by the application of a current density
to the rectangular plates;
Superconducting filaments are not modelled within the strands (strand
cross-section is considered
homogeneous);In the center: conductor with IFCC physics (on the right), compared with same conductor without special physics (on the left), mirrored over the center-line of the magnet (
no IFCC
IFCC
Model: Matthias Mentink
Slide10Study 1: Strand without imposed current
- Equivalent Magnetization Formulation (W-V)
10
Strand diameter: D
strand
={0.5, 0.7, 0.9} mm;
Imposed time-constant: =10 ms (for all simulations);Free triangular mesh + mesh refinement in strand border (150k elements, 1 min 27 s of computation time)Magnetic dipole field lines (red arrows);Changing the strand diameter, ΔB amplitude and the time constant
remain the same (
=10 ms,
Δ
B
max
=10 mT
);
Single round strand:
everything is as
expected for this particular case
.
Slide11Study 1: Strand without imposed current Is – Governing PDE
11Study 1: The governing PDE used in this case is
[3]:
where
*thanks to Lorenzo Bortot for the idea and help in model implementation.
Stokes’ theorem
Equation implemented in COMSOL
According to Wilson’s textbook [1], we expect the total current
due to the IFCCs in the strand to be
zero
*.
[3]
B. Auchmann, L. Bortot, I. C. Garcia, L. D’Angelo, H. De Gersem, M. Mentink, S. Schöps and A. Verweij,
Ad-hoc Homogenisation for Interfilament Coupling Currents
, STEAM collaboration - BMBF project, Internal report, 2019.
Slide1212
The total current
can be alternatively written as [3]:
This means that, if we assume that initial condition are such that:
and this condition will remain the same for the other analyzed cases.
Our FEM analyses show that
IFCCs are localized on the strand border and perfectly balanced as expected.
[3]
B. Auchmann, L. Bortot, I. C. Garcia, L. D’Angelo, H. De Gersem, M. Mentink, S. Schöps and A. Verweij,
Ad-hoc Homogenisation for Interfilament Coupling Currents
, STEAM collaboration - BMBF project, Internal report, 2019.
Study 1: Strand without imposed current
I
s
Slide13Study 1.2: Self-field (strand with imposed current Is
and not subjected to external field)13
Study 2: If we consider a strand with an imposed current
Is, the governing PDE is
[3]:
where
is the impressed current density and
is a winding function.
Integrating this expression along the strand’s cross-section
Ω
and using
Stoke’s theorem, we obtain
:
This equation shows a paradox
[3].
If the magnetization current (third term) should be zero and if
changes in time, the lefthandside term has to change in time as well. But this implies that
the magnetization current
cannot remain
zero
, as assumed in the previous slide.
This
paradox
comes from IFCCs formulation itself, because it is implemented only for the case in which IFCCs are due to an external varying magnetic field, and not for the strand’s own field.
[3]
B. Auchmann, L. Bortot, I. C. Garcia, L. D’Angelo, H. De Gersem, M. Mentink, S. Schöps and A. Verweij,
Ad-hoc Homogenisation for Interfilament Coupling Currents
, STEAM collaboration - BMBF project, Internal report, 2019.
Slide1414
From FEM perspective, we investigated a case in which we consider a time-varying current
[A], where
A, applied to a strand in z direction (no external varying magnetic field is applied);
The magnetic flux density
can be expressed as sum of
, generated by the transport current density
, and
, due to the magnetization current density
:
The
magnetization current
is
not zero
in this case (i.e., at t=100 ms,
A). This contradicts the divergence-free assumption of the magnetizing current. FE implementation confirms that IFCCs formulation presented by Wilson
[2]
cannot be used in the case of strand’s own field, indeed IFCCs are
not balanced
in this case.
Field
in the strand
Magnetization current density[2] M. Wilson, Superconducting magnets. Oxford University Press, 1983.
Study 1.2: Self-field (
strand with imposed current
I
s
and not subjected to external field
)
Slide1515
is the total magnetic field,
is the imposed-current-field contribution and
the IFCC-field contribution.
is not divergence-free anymore
is in the order of mT
, so we did not notice so far the inconsistency of the magnetization formulation for the case
because in our magnets the produced field is in the order of T.
Study 1.2: Self-field (
strand with imposed current
I
s
and not subjected to external field
)
Slide16Study 2: Equivalent Resistivity Formulation (M-M) for a single round strand without imposed current I
s – 1st approach (M. Mentink) 16
An alternative approach to model IFCCs is the use of an Equivalent Resistivity Formulation. We made
use
of COMSOL coil feature, which imposes the following conditions within the strand:
(coil excitation);
An equivalent electrical conductivity (in S/m) is applied to the conductive layer of thickness
:
where
is the electrical resistivity and
R
s
the strand radius.
Field amplitude and time constant are in accordance with the expectations
(
and
);
Equivalent Resistivity Formulation returns results in accordance with the Equivalent Magnetization Formulation for IFCCs modelling in single strand.
Slide1717
Total field
Applied field
Induced field
Study 3: Equivalent Resistivity
Formulation (B-B)
for a single round strand without imposed current
I
s
–
2
nd
approach (C.
Barbagallo, L. Bortot)
M-model
K-model
M-model
(Equivalent Magnetization model):
[4]
E. Ravaioli, B. Auchmann, M. Maciejewski, H.H.J. ten Kate, A.P. Verweij, “Lumped-Element Dynamic Electro-Thermal model of a superconducting magnet”, Cryogenics, vol. 80, 346-356, 2016.
K-model
(Current density model - inter-filament dissipative
loops [4])
Equivalent thickness of the strand annulus
We are supposing that IFCCs flows in the strand annulus of thickness
t
eq
and equivalent resistivity
.
COMSOL coil feature
Slide1818
M-model/K-model
J
m
is divergence-free
M-model/K-model
Study 3: Equivalent Resistivity Formulation (B-B) for a single round strand without imposed current Is – 2st approach (C. Barbagallo, L. Bortot) Magnetic field map is identical for the two models;Consistent variation field and time constant between the two models;
Divergence-free assumption of Jm is verified for both models.
Slide1919
Changing the value of time constant
(1 ms, 1s, 1000 s), the two different approaches return consistent field variation for the three investigated cases;
Magnetizing current is balanced along the strand border and could be represented as a cosinusoidal function:
)
Study 3: Equivalent Resistivity
Formulation (B-B)
for a single round strand without imposed current
I
s
–
2
st
approach (C. Barbagallo, L. Bortot)
Slide2020
From FEM perspective, we investigated a case in which we consider a piecewise-current function
[A], where
A, applied to a strand in z direction (no external varying magnetic field is applied);
The magnetic flux density
can be expressed as sum of
(self-field), generated by the transport current density
, and
(induced field), due to the magnetization current density
:
The total
magnetizing current
is
zero along the strand border
. The divergence-free assumption of the magnetizing current is verified. This IFCCs formulation could be used to simulate IFCCs also in the case of strand’s own field, indeed IFCCs result
well balanced
and, as expected, the decay of IFCCs starts once current plateau is reached.
J
m
is divergence-free
K-model
Study 3: Equivalent Resistivity
Formulation (B-B)
for a single round strand with imposed current
I
s –
2st approach (C. Barbagallo, L. Bortot)
Slide21J
m is divergence-freeK-model21
For
after 10 ms from time instant (t=1 s) in which current plateau is reached, IFCCs start decaying, as expected.
Magnetizing current is balanced along the strand border and is once again representable as cosinusoidal function:
)
Study 3: Equivalent Resistivity
Formulation (B-B)
for a single round strand with imposed current
I
s
–
2
st
approach (C. Barbagallo, L. Bortot)
Slide2222
Summary on IFCCs formulation for single strand model – Is=0
Equivalent Magnetization formulation (M. Wilson - A. Verweij):
Equivalent Resistivity formulation (M. Mentink):
Equivalent Resistivity formulation (C. Barbagallo - L. Bortot):
Equivalent Magnetization
(W-V)
Equivalent
Resistivity (
M-M
or B-B)
(W-V)
(B-B)
For a single strand subjected to an external time-varying magnetic field and without an imposed current, the three analyzed formulations return consistent results in terms of magnetic field variation and IFCC time constant.
(
M-M)
Legend:
Correct
Wrong
…
To be studied
~
Almost
Correct
Slide2323
Summary on IFCCs formulation for single strand model – Is 0
Equivalent Magnetization formulation (M. Wilson - A. Verweij):
Equivalent Resistivity formulation (M. Mentink):
Equivalent Resistivity formulation (C. Barbagallo - L. Bortot):
Equivalent Magnetization
(W-V)
Equivalent
Resistivity (
M-M
or B-B)
x
(W-V)
(
M-M)
(B-B)
For a single strand subjected to an external time-varying magnetic field and with an imposed current, only Equivalent-Resistivity-based formulations seem to reproduce consistent results to simulate IFCCs.
Legend:
Correct
Wrong
…
To be studied
~
Almost
Correct
Slide24Rutherford cable modelling24
Slide25Rutherford cable modelling
25Rutherford cable (real image)
How could we model it?
Round strands
Octagonal
strands
[5][5] C. Barbagallo, Quench Protection Heaters FE Analysis and Thermal Conductivity Measurements of Epoxy-Impregnated Nb3Sn Cables, EDMS nr. 2066640 v.1.CERNHow did we model it for IFCC physics simulations?Homogenized cable
Slide26Study 4: Stack of four Rutherford cables – Homogenized vs Multi-strand model – Equivalent Magnetization Formulation
26
Stack of four Rutherford cables submitted to an external varying magnetic field;
D
strand=0.9 mm,
(imposed in the simulation);
Homogenisation density factor
(for this case)
to compensate discretization error in the
homogenized
model;
Same field variation
amplitude
(
Δ
B=5.47 mT) but
lower time constant in the
homogenized
model
() than the multi-strand one (
). Multi-strand model
has almost the same time constant of single strand model. Multi-strand modelHomogenized model
(W-V
) formulation
(
W-V-homo) formulation
Slide27Study 4.1: Stack of four Rutherford cable –
Homogenized model - Modified IFCC equation for the homogenized model – (W-V-homo-mod)27
IFCC equation:
→
equivalent
because
Decay term
Driving term
Modified IFCC equation for
homogenized
model:
Equations were implemented in COMSOL with
Dode
(Domain ODEs and DAEs) module;
By using this approach, in which p=0.5 is a fitting
parameter for this case,
we obtained
same field variation
(
and
time constant
(
=9.6 ms) for both models;
This approach could represent a valid candidate for modelling magnets by using homogenized cables.
Case
D
strand
=0.9 mm
Homogenization + modified IFCC equation
Slide2828
Summary on IFCCs formulations for Rutherford cable stack – Is=0
Equivalent Magnetization formulation (M. Wilson - A. Verweij):
Equivalent Magnetization formulation for homogenized
models (M-V-homo):
Equivalent Magnetization
(W-V)
Equivalent
Magnetization-Homo (W-V-homo)
Modified Equivalent
Magnetization-Homo (
W-V-homo-mod)
Equivalent
Resistivity Formulation (
M-M
or B-B)
x
…
(W-V)Legend:For a stack of Rutherford cables, the (W-V) formulation applied to the multi-strand model returns consistent results with the single strand model. The (W-V-homo) formulation applied to the homogenized model gives a correct field variation amplitude, but incorrect time constant. However, this problem is fixed applying the (W-V-homo-mod) formulation to the homogenized model. CorrectWrong… To be studied
(W-V-homo)Modified Equivalent Magnetization formulation for homogenized models (M-V-homo-mod):
(W-V-homo-mod)
Slide2929
Summary on IFCCs formulations for strand and Rutherford cable
I
s
=0,
time varying B
ext appliedSingle strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo-mod)Equivalent Resistivity (M-M or B-B)
N/A
N/A
Cable
stack
x
…
I
s
0,
no B
ext
applied
Single strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo-mod)Equivalent Resistivity (M-M or B-B)xN/AN/A Cable stack……
…
…Single strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo-mod)Equivalent Resistivity (M-M or B-B)x
N/AN/A
Cable stack
…
………Legend:
Correct
Wrong
…
To be studied
~
Almost
Correct
N/A
Not Available
Slide30Conclusion (1/2) – Single round strand and Rutherford cable modelling
30 Single round strand and Rutherford cable modelling:The Equivalent Magnetization formulation (W-V) reproduces results in line with expectations
when it is applied to a single strand subjected to an
external time varying magnetic field and no current is imposed in the strand. The total magnetizing current
Im is zero as expected;
The Equivalent Magnetization formulation (W-V)
is inconsistent when a time varying current is imposed to a strand and no external magnetic field is considered. In this case the magnetizing current Im is not zero anymore;Equivalent Resistivity formulation (M-M or B-B) fixes the issue of the diverge-free of magnetizing current density. It could represent a valid candidate for IFCCs modelling in a single strand as well as in magnets. The study is still on going;Using the Equivalent Magnetization
formulation for homogenized cables (W-V-homo) in cable stacks, we obtained correct magnetic
amplitude, but an
incorrect time constant
in the homogenized model
. However, proposed
modified IFCC equation
for homogenized
cable (W-V-homo-mod)
returns values of
magnetic amplitude and time constant in line with the multi-strand model
.
Slide31Magneto-Thermal Analysis of FRESCA2 Magnet – Energy Extraction and CLIQ simulations
31
Slide32Magnet simulation overview
32Developed within a collaboration between CERN and CEA Saclay;Nb3Sn dipole magnet;
Magnetic field: 13 T;
100 mm clear bore.
Block coil design with 4 double pancake coils:
42 turns outer layer;
36 turns inner layer;Cable: 40 strands of 1 mm;Bladder and key concept for loading.FRESCA2 Nb3Sn dipole magnetImage: G. Willering
Slide33Magneto-thermal model
33Magnetoquasistatic general PDE:
Heat balance equation:
The magnetoquasistatic field solution determines the magnet’s electrodynamics and the related thermal losses
[6].
[6]
L. Bortot et al,
A 2-D Finite-Element Model for Electrothermal Transients in Accelerator Magnets
, IEEE Transaction on Magnetics, 2018, 54.3: 1-4.
Investigated cases for
EE
(
E
nergy
E
xtraction) (@I
mag
=12’108 A and
T
bath
=1.9 K,
Imag=10’000 A and Tbath=4.5 K, Imag=6’000 A and Tbath
=1.9 K):Quench all (all the turns are quenched at the same instant);
IFCCs (Inter-Filament Coupling Currents physics).Investigated cases for CLIQ (The Coupling-Loss-Induced Quench) @ Tbath = 1.9 K:Imag = 12’108 A, VCLIQ=1’250 V, C=50 mF;Imag = 5’000 A, VCLIQ=1’250 V, C=10 mF.
Slide34Homogenized model vs Multi-strand model
34
Homogenized coil*
Multi-strand coil
*built by Lorenzo Bortot
Magnetic implementation (all domains):
Magnetization
(in Am-1) is applied to superconducting cable domains;The external current density
(in Am
-2
) is applied to cables;
Magnetic insulation condition are applied to the external boundaries of the geometry;
As main result, magnet current discharge is obtained.
Thermal implementation (only superconducting coil):
(helium bath temperature), thermal insulation on external boundaries;
A
volumetric
heat source
(in Wm
-3
) is applied to the superconducting coil;
As main result, IFCC losses are obtained.
Slide35Aim of the study and roadmap – FRESCA2 magnet
35Aim of the study: Restructuring of the pre-existent homogenized model of FRESCA2 magnet;Modelling of the strands in FRESCA2 superconducting coil in order to determine the impact of homogenization on IFCC
behavior;
Simulate EE (Energy Extraction) and CLIQ (The Coupling-Loss-Induced Quench).
SIGMA
SIGMA-restructured
Magnetization formulation
Homogenized model
Multi-strand model
Magnetization formulation for homogenized
cables (W-V-homo)
Modified IFCC equation for homogenized cables
(W-V-homo-mod)
Magnetization for multi-strand
cables (W-V)
Slide36FRESCA2 homogenized model – Magneto-thermal simulation
36
In the homogenized model, cables are modelled considering homogeneous thermal properties. The model was generated by using SIGMA (S
TEAM Integrated Generator of
Magnets for Accelerator)
[7].
[7] cern.ch/STEAMSIGMA-homogenized modelSIGMA-homogenized model
Slide37FRESCA2 – SIGMA restructured model
37[8] M. Mentink, “Development of the STEAM COMSOL Magnet Simulation Tool”, CERN Technical note, EDMS no. 2054126, 2018.[9]
B. Bokharaie , “Enhanced automation of magnet model generation”, CERN Training Report, Aarhus University School of Engineering, 2019.
Starting from the existing model in SIGMA, we performed the minimum amount of changes
to the model in order to make a restructuring with the aim to
reduce computation time;Group identical properties together in common variables groups to reduce the computation time of the model [8,9].Steps to follow:Disablement and deletion of variable groups and global parameters related to single turns;Adjustments to physics engine: deletion of ISCC physics and persistent magnetization;Definition of global material properties (i.e., heat capacity of the entire turn);Definition of Jz, i.e., the direction of the electrical current through the magnet coils;
Definition of new junction boxes for magnet current;
Definition of new variable group Voltages.
Before restructuring
After restructuring
Slide38FRESCA2 Magneto-thermal simulation: comparison between SIGMA and SIGMA restructured homogenized model
38Homogenized SIGMA restructured model returns the same results of the homogenized SIGMA model for the two investigated cases (Quench all
, IFCCs);
Number of mesh elements: 5’563.
Quench
all
IFCCsHomogenized SIGMA model18 min 3 s2 h 1 min 20 sHomogenized SIGMA restructured1 min 12 s3 min 49 s
Computation time was significantly reduced after restructuring operations.
Slide39Magneto-thermal analysis results – Energy Extraction simulations39
Slide40Homogenized vs. Multi-strand model – (W-V-homo) vs. (W-V) formulations – Energy Extraction simulations
40Homogenized model: IFCC magnetization and losses are considered at a cable level, using homogenization approximation;
Multi-strand model: IFCC magnetization and losses are considered at a strand level;
How well does the homogenized approach approximate the multi-strand approach?
Equivalent Magnetization equation
H
omogenized model
(W-V-homo)
Multi-strand model
(W-V)
Homogenized model
Multi-strand model
Slide41Comparison between homogenized and multi-strand model – Magneto-thermal analysis – Energy Extraction
41IFCCs losses [W]
For considered discharge (see table), multi-strand model gives larger IFCCs
losses than the homogenized
model. In particular, the peak value is 10% higher than the homogenized model;
Multi-strand model: more IFCCs losses → large coil fraction quenches → higher quench resistance → faster
(see next few slides);These results represent a quantitative estimation of the impact of homogenization formulation on IFCC behavior.
10%
Circuit parameters
Circuit parameter
Symbol
Value
U.M.
Initial
magnet current
I
0
12,108
A
Circuit inductance
L
cir
1E-06
HEnergy extraction resistanceREE81E-03Ω
Slide42Comparison between homogenized and multi-strand model – Magneto-thermal analysis – Energy Extraction
42Temperature [K]
5%
Quench all: temperature is the same in both homogenized and multi-strand models.
IFCCs: temperature is higher of 5% in the multi-strand model than in the homogenized model.
Quench resistance [
Ω
]
7%
Quench all: quench resistance is the same in both homogenized and multi-strand models.
IFCCs: quench resistance is higher of 7% in the multi-strand model than in the homogenized model.
Slide43Comparison between homogenized and multi-strand model – Magneto-thermal analysis – Energy Extraction
43Magnet current discharge [A]
6.3%
Magnetic flux density [T]
5.3%
ZOOM
Slide44Comparison
between homogenized and multi-strand model – Magneto-thermal analysis – Energy Extraction44
Quench all: electrical resistivity of copper is the same in both homogenized and multi-strand models.
IFCCs: electrical resistivity of copper is higher of 7% in the multi-strand model than in the homogenized model.
Electrical resistivity of copper [
Ω·
m]7%Voltage at magnet terminal [V]
4%
ZOOM
Slide45Comparison between homogenized and multi-strand model – Magneto-thermal analysis – Energy Extraction
45MIITs [MA2s]
1%
Magneto-thermal
simulation results – Energy Extraction
Quench
all
(Homo vs Multi-strand)
Q
IFCC
T
peak
R
quench
I
mag
B
ave
ρ
Cu
V
peak
MIITs
NA======
=IFCCs
(Homo vs Multi-strand)↑ of 10 % in Multi-strand↑ of 5 % in Multi-strand↑ of 7 % in Multi-strand↓ of 6.3 % in Multi-strand↓ of 5.3 % in Multi-strand↑ of 7 % in Multi-strand↑ of 4 % in Multi-strand↓ of 1 % in Multi-strand
Legend:N/A Not available
↑
Higher
↑ LowerZOOM
Slide46Comparison with experimental data (EE simulations)
46Both multi-strand and homogenized models have a reasonable
match with experimental data, despite the differences in IFCCs physics implementation.
Simulation results follow the SIGMA input parameters (i.e. no parameter optimization to match the experimental results)
Slide47Final overview on magneto-thermal simulation – Homogenized model vs. Multi-strand model – Computation time
47No. of mesh elements: Homogenized model (9’357 elements), Multi-strand model (1’500’707 elements);
Computation time for the investigated cases
Computation time increases
after modelling strands in the cables.
Slide48Modified Equivalent Magnetization formulation for FRESCA2 Magnet -Energy Extraction
48
Slide49Modified Equivalent Magnetization formulations – (W-V-homo-mod) formulation – Energy Extraction
49Homogenized model current discharge curve, obtained by using modified IFCC equation for homogenized model, better approaches the curve of multi-strand model
.
Modified IFCC equation for homogenized model (p=0.5):
Equations were implemented in COMSOL with
Dode
(Domain ODEs and DAEs) module;
IFCCs
Homogenized
SIGMA
2 h 1 min 20 s
Homogenized
SIGMA restructured
3 min
49 s
Multi-strand
22 h 28 min 47 s
Homogenized
model
(modified IFCC equation - Dode)
10 min 25 sWe reduced a lot the computation time in case of EE
(Energy Extraction) with respect to multi-strand model.
Slide5050
Summary on IFCCs formulations – from strand to magnet level
I
s
=0,
time varying B
ext appliedSingle strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo-mod)Equivalent Resistivity (M-M or B-B)
N/A
N/A
Cable
stack
x
…
I
s
0,
no B
ext
applied
Single strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo-mod)Equivalent Resistivity (M-M or B-B)xN/AN/A Cable stack……
…
…FRESCA2 (EE) ~~ …Single strand modelEquivalent Magnetization
(W-V)Equivalent Magnetization-homo
(W-V-homo)
Modified Equivalent Magnetization-homo
(W-V-homo-mod)Equivalent Resistivity
(M-M or B-B)xN/AN/A
Cable stack
…
…
…
…
FRESCA2 (EE)
~
~
…
Legend:
Correct
Wrong
…
To be studied
~
Almost
Correct
FRESCA2 (EE): equivalent magnetization based formulations could be considered almost correct considering the reasonable
match with experimental data.
N/A Not Available
Slide51Magneto-thermal analysis on FRESCA2 Magnet - CLIQ protection system51
Slide52Magneto-thermal analysis with CLIQ protection system for FRESCA2 magnet – Homogenized vs. Multi-strand model
52
Homogenised
model(W-V-homo)
Multi-strand
model(W-V)All simulations were performed including IFCCs physics;No. of mesh elements: 13’876 elements (Homogenised model), 619’346 elements (Multi-strand model).Computation time: 14 min 29 s (Homogenised model), 20 h 29 min 10 s (Multi-strand model).
Slide53Magneto-thermal analysis with CLIQ: Homogenized model vs Multi-strand model – Imag=12’108 A, VCLIQ
=1’250 V, C=50 mF 53
The
slope
is 38% lower in the multi-strand model. Furthermore, after the initial fluctuation, the discharge in the two models is similar. This outcome explains why we have noticed so far a higher
than expected in all homogenized models of magnets
;Faster current decay in the multi-strand model.Losses due to IFCCs are, as consequence, lower of a factor two in the multi-strand model than in the homogenized one.
Slide5454
The slope
is 38% lower in the multi-strand model than the homogenized
model
→ faster current decay in the multi-strand model.
Faster CLIQ current decay in the homogenized model.Losses due to IFCCs are, as consequence, lower of a factor two in the multi-strand model than in the homogenized one.
Magneto-thermal analysis with CLIQ: Homogenized model vs Multi-strand model – I
mag
=5’000 A, V
CLIQ
=1’250 V, C=10 mF
Faster dampening in homogenized model
Slide55Modified Equivalent Magnetization formulation for FRESCA2 Magnet –CLIQ simulations
55
Slide56Modified Equivalent Magnetization formulations – (W-V-homo-mod)
formulation – CLIQ simulation56Modified IFCC equation for homogenized model (p=0.5):
Equations were implemented in COMSOL with
Dode (Domain ODEs and DAEs) module;
Homogenised-
dode
-model current discharge curve
approaches better the multi-strand curve only in the first part of the simulation
(until 25 ms), then the standard homogenised model curve better fits the curve of multi-strand model.
Computation time was reduced
by using modified IFCC equation –
Dode
with respect to multi-strand model.
Slide5757
Summary on IFCCs formulations – from strand to magnet level
I
s
=0,
time varying B
ext appliedSingle strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo)Equivalent Resistivity (M-M or B-B)
N/A
N/A
Cable
stack
x
…
I
s
0,
no B
ext
applied
Single strand modelEquivalent Magnetization (W-V)Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo (W-V-homo)Equivalent Resistivity (M-M or B-B)xN/AN/A Cable stack……
……
FRESCA2 (EE) ~ ~…FRESCA2 (CLIQ)To be compared to Exp. dataxx…
Single
strand model
Equivalent Magnetization
(W-V)
Equivalent Magnetization-homo (W-V-homo)Modified Equivalent Magnetization-homo
(W-V-homo)Equivalent Resistivity (
M-M
or B-B)
x
N/A
N/A
Cable stack
…
…
…
…
FRESCA2 (EE)
~
~
…
FRESCA2 (CLIQ)
To be compared to Exp. data
x
x…
Legend:CorrectWrong
…To be studied
~ Almost CorrectN/A Not Available
Slide58Conclusions (2/2) – FRESCA2 magnet
58Restructuring operations performed on FRESCA2 magnet homogenized model generated by SIGMA have permitted to reduce significantly the computation time of the magneto-thermal simulation;
Multi-strand model
was implemented in FRESCA2 magnet. In this model, the computation time increases for the high number of mesh elements required.
For EE simulations
, we noticed several differences in IFCCs simulations between homogenized and multi-strand model, due the different used formulations (i.e.,
faster current discharge in the multi-strand model).CLIQ discharge simulations implemented in the multi-strand model revealed a different slope in the magnet current curve, with respect to the homogenized model. As a consequence, IFCCs losses are quantitatively different. The current decay is faster in the multi-strand model. The CLIQ current decay is slower in the multi-strand model, especially at lower currents.Modified IFCC equation for homogenized cable (Dode-formulation) reduces the computation time with respect to the multi-strand and reasonably approaches the magnet current discharge when EE (Energy Extraction) is simulated. In case of CLIQ discharge, this formulation permits to approach better the magnet curve discharge than the standard magnetization formulation for homogenized cables, but only in the first part of the discharge.
Slide59Overall conclusions
59Extensive study was done to investigate IFCC physics at the strand, cable stack, and magnet level, with emphasis on the implications of homogenization.
Strand level, comparison
of equivalent magnetization to equivalent resistivity formulations :
Without self-field: either formulation is consistent;
With self-field, equivalent resistivity is correct.
Cable stack, comparison between homogenized conductor, modified homogenized conductor and multi-strand formulationsHomogenization results in an incorrect time-constant and a correct final field amplitude;Modified homogenization approach with fitting factor p produces consistent results with multi-strand model, with a reduced computational cost;Magnet (FRESCA2), comparison between homogenized conductor, modified homogenized conductor and multi-strand formulations:EE: Quantitative difference between homogenized and multi-strand approach;
Reasonable consistency for either approach with experimental observations;Modified homogenized approach gives greater consistency with multi-strand model;
CLIQ
:
Homogenized approach gives higher initial
dI
/
dt
, and faster
I
Cliq
dampening at lower currents, somewhat slower overall current decay;
Modified homogenized approach gives better initial consistency, but worse after 20
ms.
Slide60Thank you for your attention