Contribution to electromagnetic and thermal modelling of High Temperature Superconducting REBCO coils for protection purpose

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of High Temperature Superconducting REBCO coils for

protection purpose

Blandine Rozier

To cite this version:

Blandine Rozier. Contribution to electromagnetic and thermal modelling of High Temperature

Su-perconducting REBCO coils for protection purpose. Electric power. Université Grenoble Alpes, 2019.

English. �NNT : 2019GREAT101�. �tel-02901468�

THÈSE

Pour obtenir le grade de

DOCTEUR DE LA COMMUNAUTE UNIVERSITE

GRENOBLE ALPES

Spécialité : Génie électrique

Arrêté ministériel : 25 mai 2016

Présentée par

Blandine ROZIER

Thèse dirigée par Gérard MEUNIER

et co-encadrée par Arnaud BADEL et Brahim RAMDANE

préparée au sein du Laboratoire de Génie Electrique de

Grenoble (G2Elab)

dans l'École Doctorale Electrotechnique, Electronique,

Automatique et Traitement du Signal

Contribution to electromagnetic and

thermal modelling of High

Temperature Superconducting

REBCO coils for protection purpose

Thèse soutenue publiquement le 24 Octobre 2019,

devant le jury composé de :

M. Christophe GEUZAINE

Professeur à l’Université de Liège, Président du jury

M. David C. LARBALESTIER

Professeur à l’Université d’Etat de Floride, Rapporteur

M. Marco BRESCHI

Professeur associé à l’Université de Bologne, Rapporteur

M. Gérard MEUNIER

Directeur de recherche au CNRS, Directeur de thèse

M. Arnaud BADEL

Chargé de recherche au CNRS, Co-encadrant de thèse

M. Brahim RAMDANE

Maître de conférences à Grenoble INP, Co-encadrant de thèse

M. Pascal TIXADOR

Professeur à Grenoble INP, Invité

M. Kévin BERGER

𝜽

Final Conclusions

References

APPENDIX 1

APPENDIX 2

APPENDIX 3

APPENDIX 4

APPENDIX 5

APPENDIX 6

APPENDIX 7

I.1

I

NTRODUCTION TO SUPERCONDUCTING MATERIALS

I.1.1

D

ISCOVERY OF SUPERCONDUCTIVITY

Figure I-1 : H. Kammerling Onnes and G. Flim at the helium liquefier in Leiden laboratory (left) and the historic plot of resistance versus temperature for mercury from the 26 October 1911 experiment showing the

superconducting transition at 4.2 K (right) [3]

Figure I-2 : Comparison of field cooling experiment realized (theoritically) on an ideal conductor and a superconducting bulk

I.1.2

C

LASSIFICATION OF SUPERCONDUCTORS

I.1.2.1

T

YPE

I

/

T

YPE

II

SUPERCONDUCTIVITY

κ =

λ

L

(T)

ξ(T)

λ

ξ

κ

λ

∂𝐉

𝐬

∂t

=

1

μ

0

λ

L2

𝐄

𝐁 = −μ

0

λ

L2

∇ × 𝐉

𝐬

μ

μ

π

ξ

T

YPE

I

SUPERCONDUCTORS

κ < 1 √2

⁄

ξ

μ

μ

Figure I-3 : Magnetic field in Type I (left) and Type II (right) superconductors [8]

T

YPE

II

SUPERCONDUCTORS

2ξ

λ

Figure I-4 : Magnetic field penetration in Type II superconductors and structure of a single vortex [11]

Figure I-5 : Magneto-optical images of vortices in NbSe2 superconducting crystal at 4 K in the earth’s field

Figure I-6 : Irreversibility field H* versus temperature compared to Hc2 for various Type II superconductors

[15]

Figure I-7 : Superconducting materials as a function of their critical temperature Tc and the year of their

discovery [17], especially the cuprates in blue diamonds, widely used in engineering applications

I.1.3 E-J RELATIONSHIP IN

T

YPE

II

SUPERCONDUCTORS

I.1.3.1 F

LUX PINNING

Φ

𝐅

𝐜,𝐩𝐢𝐧

= 𝐁 × 𝐉

c

I.1.3.2 F

LUX FLOW

η

×

ρ

ρ

E = ρ

ff

(J − J

c

)

Figure I-8 : Typical E-J relationship illustrating the consequence of the flux flow concept – The point A corresponds to the critical current and as soon as J > Jc , E-J becomes linear and is characterized by the flux flow

I.1.3.3 F

LUX CREEP

R = R

0

e

−U T.k_B

Figure I-9 : Free energy function represented in 1D and probability for a flux line to jump from a pinning centre to the next one without (left) and with (right) transport current

I.1.3.4 E-J

CONSTITUTIVE LAW OF

T

YPE

II

AND ITS MODELS

C

RITICAL CURRENT

IC

μ







Figure I-10 : E-J characteristic of an HTS conductor (superconducting material + resistive matrix) and the corresponding Ic determination

T

HE CRITICAL SURFACE

T

HE CRITICAL STATE MODEL

Figure I-12 : The Critical State Model suggested by C.P Bean in 1964

𝐄(𝐉) = E

c

(

|𝐉|

J

c

(𝐁, T)

)

n(𝐁,T)

𝐉

|𝐉|

μ

Figure I-14: Jc dependency on the temperature measured on HTS samples [32]

Figure I-15: Magnetic field dependence on Jc measured on HTS tapes – Jc data are extracted from inductive

(smaller symbols) and transport (larger symbols) measurements [32]

𝐄(𝐉) = {

0 ∀ |𝐉| < J

c0

E

c

(

|𝐉| − J

c0

J

c

− J

c0

)

n0

𝐉

|𝐉|

∀ |𝐉| ≥ J

c0

Figure I-16: Comparison of the Power Law (red line) and the Alternative Power Law (blue dotted line) – Right plot is a zoom of the left plot: the Percolation model is uniformly null below 0.7

I.2

S

PECIFICITIES OF

REBCO

TAPES

I.2.1 I

C

SPECIFICITIES OF

REBCO

TAPES

I.2.1.1 A

LAYERED STRUCTURE

μ

μ

μ

μ

Figure I-18 : Structure of a 1st _{generation BSCCO wire of 2223 form (left) [43] and 2}nd _{generation REBCO}

tape (right) [44]

Figure I-19: Incident angle θ definition (left) and reduced critical current versus angle for YBCO (4 mm wide, SuperPower) tape [45] (right)

I.2.1.2 P

ERFORMANCE

I

NHOMOGENEITIES

Figure I-20: Ic scan along the length of 120 m long coated conductor [49]

θ

I

c

= f(x, T, |𝐁|, θ)

I.2.2 MECHANICAL PROPERTIES

Figure I-21: Forces and stresses in a solenoid that can lead to a delamination of the tape [52]

Figure I-22: Delaminated Fujikura coated-conductor sample after stress-strain measurements, taken from [44] (top) and delaminated REBCO tape after an experimental study, taken from [55] (bottom)

I.2.3 LARGE SCALE APPLICATIONS USING

REBCO

TAPES

Figure I-23 : (a) Superconducting transformer, Siemens [58] (b) Schematic diagram of a SMES [72] (c) Comparison of a 10 MW wind turbine [73] (d) SFCL installed in Germany (12 kV 0.8 kA), Nexans [74]

I.3

H

IGH FIELD GENERATION USING SUPERCONDUCTING

MAGNETS

I.3.1 EXISTING TECHNOLOGIES

I.3.1.1 LTS

MAGNETS

I.3.2 R&D STAGE TECHNOLOGIES

:

HTS

/

H

YBRID MAGNETS







I.3.3 P

ROTECTION OF

HTS

MAGNETS

I.3.3.1 P

ROTECTION STRATEGIES





Figure I-25: Quench protection heaters of LHC quadrupole [95]

I.3.3.3 I

NNOVATIVE WINDING TECHNIQUES TO DEAL WITH

PROTECTION ISSUES

Figure I-26: Sudden discharge tests (left) and magnetic field stability measurements (right) realized on several coils with different insulation levels, from fully insulated (coil 1) to no insulated (coil 7) and with different

τ

Figure I-27: Summary of different winding techniques used for HTS coils and their qualitative comparison of two major features of HTS coils: self-protection and charging time delay

I.4

F

ROM OBSERVATION TO PREDICTION

:

THE NEED FOR

MODELLING TOOLS

I.6

S

OLVING

PDE

WITH NUMERICAL METHODS









Figure I-28: (a) Structured and conformal mesh (b) Structured and non-conformal mesh (c) Unstructured and conformal mesh (d) Unstructured and conformal mesh – The red dotted lines highlight the

non-conformal parts of the mesh

I.6.2 STRONG FORM

/

W

EAK FORM

𝐴(∙)

𝐴(∙)

∫ 𝑨(𝒖) ∙ 𝒗

_𝛀

= ∫ 𝒇 ∙ 𝒗

_𝛀

I.6.3 DIFFERENTIAL METHODS

I.6.3.1 F

INITE

D

IFFERENCE

M

ETHOD

f(x) = ∑

f

(k)

_(a)

k!

(x − a)

k

_{+ R}

n

(x)

n k=1

𝑅

𝑛

(𝑥) = 𝑜(|𝑥 − 𝑎|

𝑛

)

|𝑥 − 𝑎|

𝑛

→ 0

𝑥 → 𝑎

I.6.3.2 F

INITE

V

OLUME

M

ETHOD

∂ρ

∂t

+ ∇ ∙ 𝐉 = g ∀(x, t) ∈ Ω × ℝ

+

ρ

∭ ∇ ∙ 𝐅 d𝑉

𝑉

= ∬ 𝐅 d𝑆

∂𝑉

∂𝑉

u

∗

_{(x) = ∑ φ}

ni

(x). u

i N

i=1

𝜑

𝑛𝑖

(𝑥)

ℋ

ℋ

I.6.4.1 B

OUNDARY

I

NTEGRAL

M

ETHODS

𝜕

𝜕

𝜕

𝜕

𝜕

I.6.4.2 V

OLUME

I

NTEGRAL

M

ETHODS

I.6.4.3 P

ARTIAL

E

LEMENT

E

QUIVALENT

C

IRCUIT METHOD

R

k

I

k

+

∂

∂t

∑ M

pk

I

k p

= U

k

R

k

= ρ ∫ j

0k2

dΩ

ck Ωck

M

pk

=

μ

0

4π

∫

j

0p

∫

j

_0k

r

pk Ωck Ωcp

dΩ

ck

dΩ

cp

U

k

= − ∫ j

0k

∙ ∇V

Ωck

dΩ

ck

ρ

Figure I-30: Equivalent circuit representation from 3D domain to electrical components (Ii is the current

through flowing through element i and Ui is the voltage drop, Ri and Li are respectively the resistance and the

self-inductance of element i and Mij is the mutual inductance between elements I and j)

I.7

E

LECTROMAGNETIC MODELLING OF SUPERCONDUCTING

I.7.1 FINITE

E

LEMENT FORMULATIONS

I.7.1.1 A-V

F

ORMULATION

𝐁 = ∇ × 𝐀 ; 𝐄 = −

∂𝐀

∂t

− ∇V

𝛁 × (

1

μ

0

𝛁 × 𝐀) + σ (

∂𝐀

∂t

+ ∇V) = 0

∇ ∙ (

∂𝐀

∂t

+ ∇V) = 0

𝜕𝑨 𝜕𝑡

⁄

σ

ρ

Figure I-31: Nonlinear convergence of a E-J (left) or J-E (right) constitutive law of an HTS material [140]

I.7.1.2 T-Φ

F

ORMULATION

∇ × 𝐓 = 𝐉 ; 𝐇 = 𝐓 − ∇Φ + 𝐓

0

∇ × 𝐄 = −

∂𝐁

∂t

∇ ∙ 𝐁 = 0

{

∇ × (𝜌 ∇ × (𝑻

+ 𝑻

0

)) +

𝜕

𝜕𝑡

(𝜇(𝑻 − ∇Φ + 𝐓

0

)) = 0

∇ ∙ (𝜇(𝐓 − ∇Φ + 𝐓

0

)) = 0

∇ ∙ 𝑩 = 0

I.7.1.3 H-

FORMULATION

∇ × (ρ ∇ × (𝐇)) + μ

0

(

∂𝐇

∂t

+

∂𝐇

0

∂t

) = 0

ρ

ρ

I.7.1.4 C

OMPARISON BETWEEN

FEM

FORMULATIONS

Table I-1: Comparisons between the A-V, T-Φ and H-formulations for modelling superconducting applications

I.7.2 MINIMIZATION OF AN ENERGY FUNCTIONAL

A

NALYSIS OF PROTECTION ISSUES IN

REBCO

COILS AND REFLECTIONS

ON THE MODELLING REQUIREMENTS

II.1

P

ROBLEM STATEMENT

II.1.1 C

ONTEXT

&

THESIS ORIENTATION

II.1.2 O

BSERVATIONS

FROM

EXPERIMENTS

AND

STUDY

REQUIREMENTS

Figure II-2: Voltage signal of a double pancake coil submitted to a current ramp [159]

𝜕 𝜕 ≠

II.2

S

TUDY DECOMPOSITION

Figure II-3 : Partition of the current ramp into two parts and predominant influence of each phenomena: Stage I predominates as long as the thermal stability is not lost, then Stage II becomes predominant

II.3

S

TAGE

I:

T

RANSIENT PHENOMENA WITH CURRENT

VARIATIONS

II.3.1 HOW MAY TRANSIENT PHENOMENA CONFUSE THE

QUENCH DETECTION IN

REBCO

COILS

?

II.3.1.1 I

NDUCTANCE OF A COIL WITH HOMOGENEOUS

J

DISTRIBUTION

Φ

Φ = L ∙ I

e = − L

dI

dt

Φ

Figure II-4 : Geometry used for the definition of the inductance L in 1D

II.3.1.2 T

HE INDUCTIVE VOLTAGE COMPONENT

E

m

=

1

2

L

coil

. I

stat

II.3.1.3 C

OMPENSATION OF THE INDUCTIVE VOLTAGE

:

EXISTING

SOLUTIONS

M

ID

-

POINT MEASUREMENT

P

ICK

-

UP COILS

M

SC−Pick up

= k √L

SC

. L

Pick up

V

D

= V

SC

−

L

SC

M

SC−Pick up

V

pick up

Figure II-6: Electrical circuit showing the voltage measurement using a pick-up coil associated to a resistive voltage divider [159]

𝑉

𝐿

= 𝐿

𝑐𝑜𝑖𝑙

(𝑑𝐼 𝑑𝑡

⁄ )

Figure II-7: VD recorded on a double pancake coil under a 4 T background magnetic field (voltage drops at

200, 300, 400 and 450 A correspond to current plateaus)

ρ

Ω

V

Cu,coil

= V

L

+ V

R

= L

dI

dt

+ RI

ρ

Ω

V

SC,coil

= L

dI

dt

Figure II-8: Theoretical voltage response of a copper coil (left) and a perfectly conducting coil (right) submitted to a constant current increase followed by a current plateau

Figure II-9: Voltage signal recorded on the small REBCO coil

II.3.2 MAGNETIZATION EFFECTS

Figure II-10: Current density distribution in a superconductor subjected to an alternative current generator for the first time after cooled down below Tc – In this case, Imax < Ic

Figure II-11: Memory effect recorded on a double pancake coil made of 12 mm-wide REBCO tape – The blue curve represents the first current ramp after cooling (1 A/s until 600 A with some plateaus in between) and

the red curve corresponds to the second current ramp (same rate but higher current target) – Sudden voltage drops to 0 V correspond to current plateaus

II.3.3 C

ONCLUSION

:

AN ELECTROMAGNETIC MODEL

II.3.3.1 A

SSUMPTIONS

θ

II.4

S

TAGE

II:

D

ISSIPATION

II.4.1 STABILITY AND QUENCH EVENT IN A

REBCO

COIL

II.4.1.1 S

TABILITY CONSIDERATIONS

λ

Figure II-13: Spectra of disturbance energy density for LTS magnets [166]

II.4.1.2 O

CCURRENCE

:

H

OW IS A THERMAL RUNAWAY TRIGGERED

IN

HTS

COILS

?

Figure II-15 : Ic values measured on a 4 mm wide SuperOx length of REBCO at 77 K, self-field (SuperOx

data)

II.4.1.3 C

OMPARISON OF QUENCH IN

LTS

AND

HTS

I

NTERPRETATION OF QUENCH FOR

HTS

ρ

II.4.2 DEFINITIONS OF THE OBJECTIVES OF THE HOT SPOT

MODEL

-

-





II.4.3 C

ONCLUSION

:

AN ELECTRO

-

THERMAL MODEL

Figure II-16 : Reduced critical current versus angle for YBCO and Bi2223 tapes[45] (left) and magnetic field dependence of Jc for REBCO tapes from Bruker HTS [180] (right)

θ

D

EVELOPMENT OF NUMERICAL TOOLS ADAPTED TO THE PROTECTION

III.1

S

OLVING THE HEAT EQUATION WITH

FEM

AND

COMPUTATION TIME IMPROVEMENTS

III.1.1 HEAT EQUATION

ρC

p

(T)

∂T

∂t

= ∇ ∙ (λ(T)∆T) + Q



ρ







λ







III.1.2 TRANSIENT

,

NONLINEAR THERMAL FORMULATION

III.1.2.1 D

OMAIN DISCRETIZATION AND WEAK FORMULATION

T = ∑ w

i N i=1

. T

i

w

j

(x

i

) = {

1 if i = j

0 if i ≠ j

𝛛

𝛛𝐭

∑ ∫

𝛀𝐰𝐢∙ 𝛒𝐂𝐩(𝐓) ∙ 𝐰𝐣 𝐝𝛀 ∙ 𝐓𝐣 𝐣

+ ∑ ∫ 𝛁(

𝐰𝐢

)

∙ 𝛀 𝛌

(

𝐓

)

∙

𝛁(

𝐰𝐣

)

𝐝𝛀 𝐣 ∙ 𝐓𝐣 =

∫

𝐰𝐢∙ 𝚪 𝛌

(

𝐓

)

∙

𝛁(

𝐓

)

∙ 𝐝𝐒

⃗

+

∫

𝐰𝐢∙ 𝛀 𝐐(𝐓) 𝐝𝛀

∫

_𝛤𝑤𝑖∙𝜆

(

𝑇

)

∙

𝛻(

𝑇

)

∙ 𝑑𝑆

⃗

=

∫

_𝛤𝑤𝑖∙𝑞0𝑑𝑆

∑ ∫ 𝛻(

_𝛺 𝑤𝑖

)

∙𝜆

(

𝑇

)

∙

∆(

𝑤𝑗

)

𝑑𝛺 ∙ 𝑇𝑗

III.1.2.2 T

IME DISCRETIZATION

(

1

∆t

n

∑ ∫

wi∙ Ω ρCp(T) ∙ wj dΩ

+ ∑ ∫ ∇(

wi

)

∙ Ω λ

(

T

)

∙

∇(

wj

)

dΩ

)

∙ Tjn =

1

∆t

n

∑ ∫

wi∙ Ω ρCp

(

T

)

∙ wj dΩ

∙

Tjn−1

+ ∫

wi∙ Γ q₀dS +

∫

wi∙ Ω Qn

(

T

)

dΩ

III.1.2.3 N

ONLINEARITIES

λ

∂ℛes

k

∂X

∙ ∆X = − ℛes

k

ℛes

_ik

= (

1

∆t

n

∑ ∫

wi∙ Ω ρCp(T) ∙ wj dΩ

+ ∑ ∫ ∇(

wi

)

∙ Ω λ

(

T

)

∙

∇(

wj

)

dΩ

)

∙ Tjk −

1

∆t

n

∑ ∫

wi∙ Ω ρCp

(

T

)

∙ wj dΩ

∙

Tjn−1

− ∫

wi∙ Γ q₀dS −

∫

wi∙ Ω Qn

(

T

)

dΩ

ℐ

_ijk

=

∂ℛes

j k

∂T

i

=

1

∆t

n

∫

wi Ω ρ

(

∆Cp ∆T

(

Tj k_{− T} j n−1

₎

_{+ C} p

)

wj dΩ

+ ∫ ∇(

wi

)

Ω

(

∆λ ∆TTj k_{+ λ}

_{) ∇(}

_w j

)

dΩ

III.1.3 TOWARDS A REDUCTION OF THE COMPUTATION TIME

III.1.3.1 N

ONLINEAR CONVERGENCE

∂y

∂t

= f(t, y(t))

y

n+1

− y

n

∆t

n+1

= f

n

y

n

− y

n−1

∆t

n

= f

n

y

n+1

= y

n

(1 +

∆t

n+1

∆t

n

) −

∆t

n+1

∆t

n

y

n−1

y

n+1

= 2y

n

− y

n−1

Figure III-2: Convergence plot of the time discretization study for a thermal runaway problem

Figure III-3 : Nonlinear iteration number per time step with and without a prediction (thermal runaway case) – Maximum temperature versus time plot (top) – Nonlinear iteration number per time step without

(middle plot) and with (bottom plot) a prediction tool for initialization

O

PTIMIZATION OF THE R ELAXATION FACTOR

α

X

k+1

_{= X}

k

_{+ α ∆X}

k

α

ℛ

α

𝛼 ∈ [0 ; 1]

𝛼 ∈ [1 ; 2]

∂ (‖ℛ

k+1

_‖

2

)

2

∂α

k

= 2 ∑ ℛ

ik+1

(∑

∂ℛ

ik+1

∂X

_jk+1

δX

j k N j=1

)

N i=1

= 0

χ

(‖ℛ

𝑘+1

_‖

2

)

2

∂χ

k+1

∂α

k

= ∑ ℛ

ik+1 N i=1

δX

_ik

= 0

Table III-2 : Number of nonlinear iteration and CPU time to reach the convergence [188] (normal NR = no relaxation - residual NR = Line Search Method – functional NR = minimization of the Energy Functional)

Figure III-4 : Averaged iteration count of Newton-Raphson method per time step using different algorithms of the optimization of the relaxation factor (with different values of external applied field)[131]

III.1.3.2 A

DAPTIVE TIME

-

STEPPING METHODS



M

ETHOD

1:

RESTRICTION ON THE STATE VARIABLE VARIATIONS

α

β

α

β

α

Figure III-5 : Flowchart of Method 1

M

ETHOD

2:

I

NTEGRAL CONTROLLER

ε

ε = ΔT

setpoint

− ∆T

computedn

∆t

n+1

= (

γ. ∆T

setpoint

∆T

_computedn

)

1 k ⁄

∆t

n

γ

δ

∆𝑇

_{𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑}𝑛

> 𝛿 ∆𝑇

𝑠𝑒𝑡𝑝𝑜𝑖𝑛𝑡

C

OMPARISON

∆𝑇

𝑚𝑎𝑥

= 0.1 𝐾

α

β

δ

Figure III-9: Step size sequence (top) and successive values of ΔTcomputed (bottom) – Method 2

Figure III-10: Tmax time evolution computed with adaptive method 2 (red markers) compared to the

reference solution computed with a fix time step of 20 ms (solid blue line)

III.1.3.3 C

ONCLUSION ABOUT COMPUTATIONAL IMPROVEMENTS

III.2

T

RANSIENT PHENOMENA ANALYSIS USING A VOLUME

INTEGRAL METHOD

III.2.1.1 F

ORMULATION ORIGINALITY





III.2.1.2 A

DVANTAGES AND DRAWBACKS OF THE

J-

FORMULATION

CONSIDERING A SUPERCONDUCTING PROBLEM

M

ESH REDUCTION

C

URRENT DENSITY CONSERVATION

ρ

I(Γ) = ∫ 𝐉 ∙ 𝐧 dΓ

Γ

Figure III-12 : Current density distribution computed with the J-formulation (left, GMSH view) and the V-formulation (right, Flux® view)

Table III-4 : Incoming and outgoing current density fluxes computed for each solution

Figure III-13 : Example of a 2D geometry connected to an external electric source (current or voltage source)

F

ULL MATRIX GENERATION

III.2.2.1 C

ONTINUOUS EQUATIONS IN

3D

∇ ∙ 𝐉 = 0

𝐄 = −

∂𝐀

∂t

− 𝛁V

𝐄 = ρ

c

(𝐉)𝐉

ρ

𝐀 = 𝐀

𝟎

+ 𝐀

𝐬𝐜

(𝐉)

𝐀

𝐬𝐜

(Q)

P

= μ

0

∫ G

3D

(PQ) ∙ 𝐉 dΩ

c Ωc

= μ

0

∫ G

3D

(r) ∙ 𝐉 dΩ

c Ωc

G

3D

(r) =

1

4π ∙ r

ρ

𝑐

(𝐉) 𝐉 + μ

0

∂

∂t

∫ G

_Ωc 3D

(r) ∙ 𝐉 dΩ

c

= −

∂𝐀

𝟎

∂t

− ∇V



∇ ∙ 𝑱 = 0



𝑱 ∙ 𝒏 = 0

III.2.2.2 D

ISCRETIZATION AND CIRCUIT COUPLING

𝐉 = ∑ 𝐰

𝐟𝐤

I

k 𝐤

𝐰

𝐟𝐤

∙ 𝐧 = {

±

1

S

k

0

∇ ∙ (𝐰

𝐟𝐤

) = ±

1

V

e

Figure III-14 : Representation of first order facet shape functions for a tetrahedron (left) and a hexahedron (right) reference element [203]

([R] +

∂

∂t

[L]) [I

B

] =

∂

∂t

[A

0

] + [∆V]

R

ij

= ∫ 𝐰

𝐟𝐢 Ω_i

∙ ρ ∙ 𝐰

𝐟𝐣

dΩ

i

L

ij

=

μ

0

4π

∫ 𝐰

fi

∫

𝐰

𝐟𝐣

r

dΩ

c

dΩ

i Ωc Ωi

A

0i

= − ∫ 𝐰

𝐟𝐢

∙ 𝐀

𝟎

dΩ

i Ωi

∆V

i

= − ∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

i Ωi











∆V

i

= − ∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

Ωc

= − (∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

Ω1

+ ∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

Ω2

)

∆V

i

= − (∫ 𝐰

𝐟𝐢

∙ 𝐧

𝟏

V dS

f Sf

− ∫ ∇ ∙

Ω1

𝐰

𝐟𝐢

V dΩ

c

+ ∫ 𝐰

𝐟𝐢

∙ 𝐧

𝟐

V dS

f Sf

− ∫ ∇ ∙

Ω2

𝐰

𝐟𝐢

V dΩ

c

)

∆V

i

= − (

1

S

f

∫ V dS

f s_f

−

1

𝒱

e1

∫ V dΩ

c Ω₁

−

1

S

f

∫ V dS

f s_f

+

1

𝒱

e2

∫ V dΩ

c Ω₂

)

∆V

i

= −(V

fi

− V

e1

− V

fi

+ V

e2

) = V

e1

− V

e2

𝒘

𝒇𝒊

∙ 𝒏 = 0

∆V

i

= − ∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

Ω_c

= − ∫ 𝐰

𝐟𝐢

∙ ∇V dΩ

Ω₁

∆V

i

= − (∫ 𝐰

𝐟𝐢

∙ 𝐧

𝟏

V dS

f Sf

− ∫ ∇ ∙

Ω1

𝐰

𝐟𝐢

V dΩ

c

)

∆V

i

= − (

1

S

f

∫ V dS

f s_f

−

1

𝒱

e1

∫ V dΩ

c Ω₁

) = V

e1

− V

fi

Figure III-16: Physical interpretation of ΔVi in case of an external facet – Here, the energy source is

connected to element 1 (among others) via facet i (so Vfi = Vs), dotted lines represent the rest of the mesh, but a

focus is made on element 1

Figure III-17 : Schematic view of the circuit generation from the dual mesh of a 3D domain Ωc : (a) 3D

domain where barycentres of elements 1, 2, 3 and 4 are represented in red while barycentres of facets 5 and 6 (connected to an external source) are represented in black – (b) Equivalent electrical circuit generated from the dual mesh with Zk representing the impedance of branch n°k – (c) Details of an equivalent branch of the circuit

[M][∆V] = [0]

[M]

t

[I

L

] = [I

B

]





[M] ([R] +

∂

∂t

[L]) [M]

t

_[I

L

] = [M] (

∂

∂t

[A

0

] + [∆V

])

III.2.2.3 E

XTERNAL ELECTRICAL SOURCE MANAGEMENT

[M

_RL

M

S

] [

∆V

_in

∆V

S

] = [0]

[∆V

_in

] = ([R] +

∂

∂t

[L]) [I

B

] −

∂

∂t

[A

0

]

[M

_RL

] ([R] +

∂

∂t

[L]) [M

RL

]

t

_[I

L

] = [M

RL

]

∂

∂t

[A

0

] − [M

S

][∆V

S

]

C

URRENT DRIVEN DEVICE

∫ 𝐉 dS

in Sin

= I

S

& ∫

𝐉 dS

out Sout

= −I

S

∑ ∫ 𝐰

𝐟𝐩

I

p

dS

in Sin p

= I

S

& ∑ ∫

𝐰

𝐟𝐩

I

p

dS

out Sout p

= −I

S

[M

S

]

t

[I

L

] = [I

s

]

(

[M

RL

] ([R] +

∂

∂t

[L]) [M

RL

]

t

_[M

S

]

[M

S

]

t

[0]

) ∙ (

_[∆V

[I

L

]

S

]

) = (

[M

RL

]

∂

∂t

[A

0

]

[I

S

]

)

III.2.3 ADAPTATION TO

HTS

COILS

III.2.3.1 F

ROM

3D

TO

2D

θ θ

θ

𝐉 = J(r, z)𝐮

θ

𝐀

= A

(r, z)𝐮

θ

G

2Daxi

= ∫ G

3D

∙ cos(θ) ∙ r dθ

2π

0

𝒞

Figure III-18 : Integration of the 3D Green kernel over the azimuthal direction

G

2Daxi

=

K

4πR

((2 − k

2

_{) ∙ J}

1

(k) − 2J

2

(k))

K

2

= (r + R)

2

+ h²

k

2

=

4rR

K²

J

1

(k) = ∫

1

√1 − k²sin²(φ)

dφ

π 2 ⁄ 0

J

2

(k) = ∫

√1 − k²sin²(φ) dφ

π 2 ⁄ 0

R

ij

= ∫ 𝐰

𝐟𝐢 S

∙ ρ(𝐉) ∙ 𝐰

𝐟𝐣

∙ 2πr

dS

L

ij

= μ

0

∫ 𝐰

𝐟𝐢

∫ G

2Daxi

∙ 𝐰

𝐟𝐣

dS

2πr

dS

S′ S

A

0i

= − ∫ 𝐰

𝐟𝐢

∙ 𝐀

𝟎

∙ 2πr

dS

S

∆V

Si

= − ∫ 𝐰

𝐟𝐢

∙ ∆V

S

∙ 2πr

dS

S

III.2.3.2 T

IME DISCRETIZATION A ND NONLINEAR RESISTIVITY

𝜕𝑅𝑒𝑠𝑘 𝜕𝐼_𝐿

𝑅𝑒𝑠

𝑘

Res

k

_{= (}

[M

RL

] ([R] +

1 ∆t

[L]) [M

RL

]

t

_[M

s

]

[M

s

]

t

[0]

) ∙ (

[I

L

]

k

[∆V

_S0

]

k

)

− (

[M

RL

] (

1 ∆t

[L]) [M

RL

]

t

_[0]

[0]

[Id]

) ∙ (

[I

L

]

n

[I

0

]

n+1

)

− (

[M

RL

]

1 ∆t

([A

0

]

n+1

_{− [A}

0

]

n

)

[0]

)

∂Res

k

∂I

L

= (

[M

RL

] ([R] + [J]

k

₍

∂[R]

∂[J]

t

)

+

1

∆t

[L]) [M

RL

]

t

_[M

s

]

[M

s

]

t

[0]

)

([𝑅] + [𝐽]

𝑘

(

_𝜕[𝐽]𝜕[𝑅]_𝑡

))

ρ

III.2.3.3 I

MPLEMENTATION

III.3

C

ONCLUSION

T

HE

J-

FORMULATION APPLIED TO SUPERCONDUCTING PROBLEMS

:

VALIDATION AND ANALYSIS

IV.1

P

RACTICAL IMPLEMENTATION OF THE MODEL

IV.2

V

ALIDATION OF THE

J-

FORMULATION BY COMPARISON

IV.2.1 EXTERNAL MAGNETIC FIELD PROBLEM

:

HTS

BULK

MAGNETIZATION

IV.2.1.1 P

ROBLEM DESCRIPTION

G

ENERAL CONSIDERATIONS

Figure IV-1: Geometry of the magnetization problem







M

ESH AND MAGNETIC FIELD SOURCE MANAGEMENT

θ

Figure IV-3: Normalized current density (J/Jc) distribution inside the bulk computed at t = 5 s (top left),

t = 10 s (top right) and t = 15 s (bottom) with the J-formulation

Figure IV-4: Induction B distribution inside the bulk computed at t = 5 s (top left), t = 10 s (top right) and t = 15 s (bottom) with the J-formulation

Figure IV-5: Trapped magnetic along the radius computed 2 mm above the top surface of the bulk (z = 7 mm) – Comparison of the J-formulation (VIM) with the H-formulation (FEM – reference)

≈

IV.2.2 TRANSPORT CURRENT PROBLEM

:

AC

LOSSES IN AN

HTS

PANCAKE COIL

IV.2.2.1 P

ROBLEM DESCRIPTION

G

ENERAL CONSIDERATIONS

μ

μ

Figure IV-6: Representation of a single pancake REBCO coil in 3D (left) and in 2D axisymmetric (right)

p

AC

(t) = ∫ 𝐄 ∙ 𝐉 dV

V

P

AC

=

1

T

∫ p

AC

(t)

T 0

dt

Figure IV-7: Current density distribution inside the turns computed at different times (cross-section view) – The AC losses computation is made from t = 5 ms, when the permanent state has been reached to remove the

M

ESH AND CURRENT SOURCE MANAGEM ENT

μ

I

s

(t) = ∫ J

θ

dS

S

θ

IV.2.2.2 AC

LOSSES VALIDATION AND MESH SENSITIVITY

Figure IV-8: Current density distribution at t = 5 s and Is = Imax computed on a coarse mesh (left) and on a

finer mesh (right) – The geometry is not to scale, the turn thickness has been enlarged for more clarity

Figure IV-9: Validation based on PAC computation and convergence plot of the problem solved with VIM -

Top plot shows the AC losses values computed for each mesh and bottom plot shows the relative error (with respect to the reference value calculated with FEM) as function of the mesh density

C

ONVERGENCE PLOT AND DISCUSSION

Figure IV-10: Computation time versus the number of elements per turn (left plot) and the error (relatively to the FEM result) versus the computation time (right plot)

IV.2.2.3 C

OMPARISON TO BENCHMARK

#3

RESULTS AT LOCAL

SCALE

Figure IV-11: Current density distribution along the width of the 11th_{, 15}th_{and 20}th _{turns for I}

s = 150 A and

t = 5 ms (first positive peak) – Comparison between FEM (from Benchmark#3) and VIM – The non-zero value at the centre of the tape is due the surface elements used in the numerical computation

Figure IV-12: Transverse magnetic field computed along the width of the 11th_{, 15}th_{and 20}th _{turns for I} s =

Figure IV-13 : Current density distribution along the width of the 11th _{turn between 0 and 5 ms (I}

s between

0 and 150 A) – The current density penetrates from the edges towards the centre, following the trend predicted by the CSM

IV.3

D

ESCRIPTION OF THE

REBCO

INSULATED COILS

MODEL

IV.3.1 DOMAIN DEFINITION

IV.3.1.1 G

EOMETRY

IV.3.2 MATERIAL PROPERTIES

IV.3.2.1 E-J

LAW

μ

ρ

ρ

SC

(|𝐉|) =

E

c

J

c

(T, 𝐁)

(

|𝐉|

J

c

(T, 𝐁)

)

n(T,𝐁)−1

𝜕ρ

SC

𝜕|𝐉|

= (n(T, 𝐁) − 1)

E

c

J

c

(T, 𝐁)²

(

|𝐉|

J

c

(T, 𝐁)

)

n(T,𝐁)−2

IV.3.2.2 J

C

DEPENDENCIES

Figure IV-14: Jc measurements in function of the magnetic field [33]

Figure IV-15: Lift factors of SuperOx 2G HTS wires measured at 4.2 K at different laboratories: University of Geneva, Lebedev Institute, Brookhaven National Laboratory (BNL), National High Magnetic Field Laboratory (NHMFL), Paul Scherrer Institute (PSI) and National Research Council (CNR-SPIN, Salemo) - All samples are from

different production runs from 2013 to 2015 [217]

IV.4

S

ENSITIVITY ANALYSIS

IV.4.1 TEST CASE PRESENTATION

IV.4.2 SENSITIVITY TO N

-

VALUE VARIATIONS

θ

θ

Figure IV-16: V-I curves computed with the model for different n-values (Jc(4.2K,B,θ)ref)

IV.4.3 SENSITIVITY TO

J

C

(B,𝜽)

VARIATIONS

I

c

(T

op

, B, θ)

_tape

=

I

c

(T

op

, B, θ)

_sample

I

c

(77K, sf)

sample

∙ I

c

(77K, sf)

tape

= LF(T

op

, B, θ) ∙ I

c

(77K, sf)

tape

θ

θ

α

α

α

α

α

≈

IV.4.4 DISCUSSION

α

Figure IV-18: Sensitivity of the output voltage on the n-value (left) and Jc(B,θ) map (right) (same scale on

IV.5

M

ODEL VALIDATION BY COMPARISON TO EXPERIMENTS

IV.5.1 DESCRIPTION OF THE

REBCO

COIL

IV.5.1.2 D

ESIGN AND FABRICATION

± μ μ ± μ ± μ μ μ

Figure IV-19: REBCO insulated coil mounted on the winding machine (left) – Top view (right)

Figure IV-20 : Electrical circuit

IV.5.3 COMPARISON BETWEEN SIMULATION AND EXPERIMENTS

IV.5.3.1 S

MALL

-

SCALE

REBCO

COIL

S

CENARIO

:

SUCCESSIVE CURRENT RAMPS

Figure IV-22: Current profile of interest for the validation

JC

SENSITIVITY AND CALIBRATION

Figure IV-23: V-I characteristic of the REBCO coil – Comparison between experimental data (solid lines) and simulation results (dotted and broken lines) – First simulation corresponds to Jc and second simulation to

±

S

UCCESSIVE CURRENT RAMPS

:

COMPARISON WITH EXPERIMENTS











Figure IV-24: Comparison between experimental data recorded on the REBCO coil and the model results – Successive charge/discharge cycles (Validation 1 up to 300 A)

Figure IV-25: Comparison between experimental data recorded on the REBCO coil and the model results – Successive charge/discharge cycles (Validation 2 up to 500 A)

Figure IV-26 : Voltage-current characteristic of a double pancake coil ramped up at 1 A/s - Comparison between experimental data (solid line) and simulation results (broken and dotted lines)

S

TUDY OF THE TRANSIENT BEHAVIOUR OF

REBCO

COILS IN

DC

CONDITIONS

V.1

A

NALYSIS OF THE VOLTAGE DRIFT WITH CURRENT

VARIATIONS

V.1.1 C

ONCEPT OF VARIABLE INDUCTANCE

e = −

∂Φ

∂t

= (

∂Φ

∂I

)

t

dI

dt

− (

∂Φ

∂t

)

I

L

SC

=

∂Φ

∂I

V.1.2 Φ-I RELATIONSHIP

Φ

n,

Φ

n

contributions: Φ = ∑ Φ

_n.

Φ

n

= ∫ 𝐁 ∙ d𝐒

Sn

𝜕

𝜕

Figure V-1: Simulated flux linkage Φ versus current I (left) and calculated inductance by substituting the Φ-I curve in Eq. (4)

V.1.3

I

NFLUENCE OF THE MAGNETIC FIELD ON THE CURRENT

DENSITY DISTRIBUTION

Figure V-2: V-I curve (left) and current density distribution over the cross section (right) of the REBCO coil submitted to a constant current rise – The superconducting layer thickness has been artificially enlarged for

Figure V-3: Magnetic flux density map generated by the REBCO coil at Is = 250 A (top) and Is = 1000 A

(bottom) – Transverse component Br (left), parallel component Bz (middle) and magnitude |B| (right)

Figure V-4: J distribution across the width of the 15th _{turns of the REBCO coil – Normalized values (right)}

and absolute values (left)

V.1.4 INTERPRETATION IN TERMS OF

S

CREENING

C

URRENT

I

NDUCED

F

IELD

V.1.4.1 S

CREENING

C

URRENT DEFINITION

Figure V-5: Local screening currents induced in a flat REBCO tape by an external field [230]

J

TOT

= J

S

+ J

SC

𝐽

𝑠

= 𝐼

𝑠

⁄

𝑆

𝑠𝑢𝑝𝑒𝑟

Figure V-6: Current density decomposition into a homogeneous component and a SCIF component for a transport current Is of 500 A

V.1.4.2 I

MPACT OF

SCIF

B

r,TOT

= B

r,S

+ B

r,SCIF

B

z,TOT

= B

z,S

+ B

z,SCIF

Figure V-7: Induction map computed for several current density distributions: a homogeneous transport current distribution Js (left), the SCIF distribution JSCIF (middle) and their sum JTOT (right) – The coil is represented

by the black rectangle - Scales are different for each quantity (Br, Bz and |B|) but is the same for the three

Figure V-8: Decomposition of the total magnetic field (blue curve) calculated with the model into a linear component related to Js (green curve) and the screening-currents field (orange curve)

V.2

A

NALYSIS OF TRANSIENT PHENOMENA AND LOSSES IN A

REBCO

COIL

Figure V-9: Current profile used for the scenario highlighting magnetization impact on the voltage of a REBCO coil











V.2.1 FIRST

/S

ECOND MAGNETIZATION

Figure V-10: Voltage response to a first and a second current ramp at the same rate of a REBCO coil (the voltage scale starts at 100 μV for clarity)

Figure V-11: Comparison of the voltage computed for the first part of the scenario (blue solid line) to the voltage of a single ramp (red dashed line)

① ②

③ ④

Figure V-12: Normalized current density distribution displayed at time 1, 2, 3 and 4 according to the notation of Figure V-11

Figure V-13 : Successive current ramps up to 700 A (dotted lines) after a first ramp up to 800 A (red solid line)

V.2.1.2 AC

LOSSES

P

AC

= ∫ 𝐄 ∙ 𝐉 dV

V

= E

c

∫ (

J

J

c

)

n

∙ J dV

V

Figure V-14: Time-evolution of the transport current Is, the voltage V and the magnetization losses Pmag –

The voltage scale starts at 100 μV for clarity

μ

Figure V-15: Total voltage repartition between an inductive signal VL and a dissipative signal VD (negative

Figure V-16: Normalized repartition of each component VL and VD of the total voltage

δ

②

δ

≈

❸,

❶,

≈

❹

δ

≈

❺

❶

❷

❸

,

δ

❺

δ

δ

②

③

①

④

,

δ ≈

⑤

Figure V-17: Filling factor δ = J/Jc versus transport current Is of two elements of the mesh – Top plots

V.2.1.3 A

N HYSTERETIC BEHAVIOUR AT THE LOCAL SCALE

V.2.2 R

ELAXATION

Figure V-18: Normalized current density distribution over the cross-section of the coil at the beginning (left) and at the end (right) of the current plateau where Is = 700 A

Figure V-19: Normalized current density distribution along the width of the tape plotted at turn n° 11 (black rectangle in Figure V-18)

Figure V-20: Magnetic field drift with time at the centre of the pancake

Figure V-21: Voltage evolution during current plateaus – Record from experiment carried out on a double pancake coil made of 12 mm REBCO tape, courtesy of Jérémie Ciceron (left) and simulation results (right)

Figure V-22: Time evolution of the AC losses

V.2.3 C

ONCLUSION REGARDING PROTECTION

Figure V-24 : Voltage signal recorded during tests of a doule pancake coil (same as Figure II-11)

V.3.2 E

XAMPLE OF A SMALL SOLENOID MAGNET

Figure V-25: 3D representation of the solenoid where each pancake are series-connected and its equivalent representation (cross-section view) in 2D axisymmetric

Figure V-26: Current profile (top) and corresponding normalized current density distribution (bottom) of the solenoid – The thickness of each turn has been widened for clarity

Figure V-27: Induction maps generated by the solenoid for a current transport Is = 500 A and decomposed

according to Eq. (7) and (8) - Scales are different for each quantity (Br, Bz and |B|) but is the same for the three

distributions of a given quantity (for instance Br,TOT, Br,s and Br,SCIF)

Figure V-28: Total voltage of the solenoid VTOT divided into an inductive component VL and a dissipative

Q

UENCH PROPAGATION INSIDE AN INSULATED COMPACT

REBCO

WINDING

VI.1

M

ODELLING APPROACH

VI.1.1 MAGNETIC FIELD DISTRIBUTION

θ

I

c

(n, I

s

) =

∫I

c

(r = r

n

, z)|

B(Is)

dz

Figure VI-1: (a) 2D axisymmetric representation of the magnetostatic problem, cross-section view and (b) integration of the B-dependency of Ic in the final model

VI.1.2 CURRENT DISTRIBUTION

VI.1.2.1 E

LECTRICAL CIRCUIT OF THE

{SC

COIL

–

PROTECTION

}

SYSTEM

Figure VI-2: Electrical circuit of a dump resistor active protection system (same as Figure I-24)

𝜏

𝑅𝐿

τ

RL

=

L

SC

(r

SC

+ R

dump

)

VI.1.2.2 T

APE MODELLING AND DI SCRETIZATION

ρ

SC

(𝐉, 𝐁, T) = {

E

c

J

c

(𝐁, T)

(

|𝐉|

J

c

(𝐁, T)

)

n−1

T ≤ T

c

ρ

N

(|𝐁|, T) T > T

c

D

ISCRETIZATION

O

RIGIN OF

I

C

VARIATIONS ALONG THE LENGTH

μ

μ

Figure VI-4: 2D characterization of a 12 mm wide REBCO tape from SuperOx using the RTR SHPM method and the corresponding 1D Ic variations at 77 K in self-field operation (left) and under 1 T transverse field (right) –

The measurements have been obtained on two different apparati (the self-field characterization has a better spatial resolution)

I

c

=

w

w − 0.6λ

cut−off

∫

|J

_x

|dy

+∞ −∞

λ

VI.1.2.3 C

URRENT DISTRIBUTION AT BLOCK SCALE

S

ELECTION OF A CURRENT DISTRIBUTION MODEL

P

block

(n) = ∫ E(n) ∙ J dV

V

Figure VI-5: Different current distribution approaches to model a single block of conductor following [179] (Atot represents the area of the total section and α is the fraction of stabilizer of the conductor

cross-section)

T

HE

C

URRENT

S

HARING ALGORITHM

I

s

= I

SC

+ I

stab

U

stab

= R

stab

(𝐁, T)I

stab

= R

stab

(𝐁, T)(I

s

− I

SC

)

U

SC

= R

SC

(I

SC

, 𝐁, T)I

SC

R

SC

(I

SC

, 𝐁, T)I

SC

+ R

stab

(𝐁, T)I

SC

− R

stab

(𝐁, T)I

s

= 0

β

n

+

(I

c Block

(𝐁, T))

n

I

sn−1

Lg

Block

E

c

R

stab

(𝐁, T)β −

(I

c Block

(𝐁, T))

n

I

sn−1

Lg

Block

E

c

R

stab

(𝐁, T) = 0

β

Q

n

= R

SC

(𝐉

SC

, 𝐁, T) ∙ (β ∙ I

s

)

2

+ R

stab

(𝐁, T) ∙ ((1 − β) ∙ I

s

)

2

VI.1.3 THERMAL PROPAGATION

VI.1.3.1 D

OMAIN DEFINITION AND BOUNDARY CONDITIONS

Figure VI-6: (a) 2D geometry of a REBCO pancake viewed from above and (b) focus on a reduced angular portion

VI.1.3.2 T

HERMAL PROPERTIES AND TAPE HOMOGENIZATION

𝐶

𝑝𝑣

= 𝜌

𝑚

∙ 𝐶

𝑝

C

pv

(T) =

∑ e

i i

∙ C

pv,i

(T)

∑ e

_{i i}

=

∑ e

i i

∙ ρ

m,i

C

p,i

(T)

∑ e

_{i i}

λ

∥

𝜆

⊥

λ

∥

=

∑ e

i _i

∙ λ

i

(T)

∑ e

_{i i}

λ

⊥

=

∑ e

_{i i}

∑

e

i

λ

i

(T)

i

ρ

λ

Figure VI-7: Typical cross section of a REBCO tape and analogy between thermal and electric conduction to outline the anisotropic behaviour of the layered structure

Figure VI-9: Equivalent heat capacity of the tape

𝜏

⊥

𝜏

∥

D(T) =

λ(T)

ρ

m

C

p

(T)

=

λ(T)

C

pv

(T)

τ

⊥

(T) =

e²

D

⊥

(T)

=

e² ∙ ρ

m

C

p

(T)

λ(T)

τ

∥

(T) =

D

∥

(T)

v

_DZ2

=

λ(T)

ρ

m

C

p

(T) ∙ v

DZ2

HOMOGENIZATION WITH THE SURROUNDING INSULATION LAYER

T=4.2K T=30K T=72K T=77K T=86K

𝛕⊥[MS] 2 17.3 41.9 43.7 46.4

𝛕∥[MS] 7.1 16.2 22.6 40.9 214.4

𝛕∥/𝛕⊥ 3.5 0.94 0.54 0.93 4.6

Table VI-1: Typical time constant values of heat propagation in the longitudinal and transverse directions – Homogenization with the insulating layer

HOMOGENIZATION WITHOUT THE SURROUNDING INSULATION LAYER

T=4.2K T=30K T=72K T=77K T=86K

𝛕⊥[MS] 0.068 0.375 1.7 1.8 2

𝛕∥[MS] 7.6 31.7 29.2 52.4 269

𝛕∥/𝛕⊥ 110 84.5 17.12 28.4 131.6

Table VI-2: Typical time constant values of heat propagation in the longitudinal and transverse directions – Homogenization with the conducting layers of the tape only

VI.1.4 COUPLING STRATEGY AND IMPLEMENTATION

λ