Heat transfer in electrical insulation of LHC cables cooled with superfluid helium

Heat transfer in electrical insulation of LHC cables cooled withsuper¯uid helium

C. Meuris a,*, B. Baudouy a, D. Leroy b, B. Szeless c

a Service des Techniques de Cryog�enie et de Magn�etisme, Direction des Sciences de la Mati�ere, CEA/Saclay, 91191 Gif sur Yvette, Franceb LHC Division, CERN, 1211 Geneva 23, Switzerland

c EP Division, CERN, 1211 Geneva 23, Switzerland

Received 23 August 1999; accepted 22 September 1999

Abstract

The electrical insulation of the Large Hadron Collider (LHC) cables constitutes a thermal barrier between the conductor and the

super¯uid helium bath. This can prevent removal of the heat dissipated in the cable by the current rise in the dipoles or by the beam

losses. The main experimental results, obtained with stacks of insulated conductors representing a piece of the actual coil, are given.

The mock-ups vary only by the material composition and the structure of the electrical insulation. Analysis of the temperature

distribution measured in the conductors as a function of the dissipated heat power makes it possible to determine the dominant heat

transfer mode in each type of tested insulation and to classify these according to their permeability to super¯uid helium. Thermal

numerical modelling of the experimental mock-ups clari®es the heat transfer path in the complex structure of the insulation and

enables calculating values of the thermal quantities characteristic of each insulation. The results of these studies have led to the

choice of the cable insulation of the LHC magnets. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Super¯uid helium; Heat transfer; Superconducting cables

1. Introduction

The temperature rise in a Large Hadron Collider(LHC) dipole conductor subjected to the dissipation ofpower due to constant beam losses (104 W/m3 in themost exposed cable when 107 protons per second hit thebeam shield) must not exceed 1.1 K. This temperatureincrease, added to the one coming from the other powersources, will balance the temperature margin of 1.4 K,di�erence of temperature between the operating tem-perature (1.9 K) and the cable critical temperature at theoperating ®eld (3.3 K) [1,2]. The conductor temperatureis determined by the equilibrium between the powerdeposited in a practically stationary regime by theselosses and the power removed by the cooling system.The latter is essentially limited by the electrical insula-tion of the cable, which, if it were perfectly tight wouldlead by thermal conduction to a temperature rise abovethe temperature margin [3].

The insulation consists of a primary wrapping en-suring the electrical insulation and a secondary wrap-ping providing mechanical adherence of the coil aftercuring. The former is made up of a polyimide tapewound around the cable with 50% overlap (KaptonÒHN type or KaptonÒ base underlaid with thin dry glasstissue). The secondary wrapping is an adhesivetape:epoxy resin-impregnated ®breglass tape or polyi-mide tape clad with a polyimide adhesive (KaptonÒLCI type), wound with a spacing, as shown in Fig. 1(a),or alternated with a dry KevlarÒ/®breglass tape, asshown in Fig. 1(b).

Numerical modelling cannot be made a priori due tolack of knowledge of the helium distribution in the cableand the compressed insulation. An experimental pro-gramme has therefore been carried out. The main ob-jective of the experimental work described below is toidentify the relative weights of the various parameterscharacterising the insulation and involved in the heattransfer, and thus in the conductor temperature rise, suchas helium volume in the cable, used materials and tapethicknesses, number of layers in the primary insulationwrapping, width of secondary insulation tape spacing,manufacturing process (polymerisation pressure, etc.).

Cryogenics 39 (1999) 921±931

* Corresponding author. Tel.: +33-1-6908-4733; fax: +33-1-6908-

4950.

E-mail address: [email protected] (C. Meuris).

0011-2275/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 1 1 - 2 2 7 5 ( 9 9 ) 0 0 1 1 5 - 0

2. Experimental mock-ups

The superconducting cables are simulated by 2.5 mmthick, 17 mm wide and 150 mm long stainless steelplates, so as to simulate the beam losses by Joule e�ectin the conductor volume. The surface of the plates ismachined to reproduce the geometry of the cablestrands. The ends are not machined in order to ensurethe tightness of the insulating tapes and thus prevent anyhelium leak. In addition, the low heat conductivity ofstainless steel makes it possible to avoid end e�ects dueto longitudinal conduction.

Allen Bradley type temperature sensors (10 X/mKaround 1.9 K) are inserted in cavities specially machinedin the plates before being embedded in epoxy resin. Twocurrent leads are welded to the ends of each conductorso as to dissipate up to 150 mW power, i.e., 1.25 W/m ofconductor. Let us notice that the losses of 104 W/m3

mentioned in the introduction correspond to 64 mW inthe experimental arrangement. The temperature sensorsmeasure the surface temperature of the conductor at themiddle of the large face, which we will designate in thefollowing as ``conductor temperature'' [4].

The sample consists of a stack of ®ve model conduc-tors, insulated and polymerised under temperature andpressure conditions comparable to those of the dipolewinding (P� 50 to 150 MPa and h� 130°C to 170°Cdepending on the material used for secondary wrapping

[5]). The sample is then inserted in a clamping systemproviding a compressive force on the stack (�60 MPa)during the tests in a pressurised He II cryostat.

The in situ acquisition and calibration system for thetemperature sensors, using standard germanium detec-tors, makes it possible to measure the temperature dif-ferences between the conductors and the helium bathwith a precision of �1 mK [6]. The conductors areheated individually or collectively by passing a currentadjustable from 0 to 10 A. Their temperatures aremeasured after establishing a stationary regime. Theexternal sample temperature, i.e. the He II bath tem-perature �1:7 K < Tb < 2:16 K�, is controlled and con-stant for all the power range.

3. Phenomenological description of typical results

Examination of all the results shows two families ofthermal behaviour. The temperature rise, DT, of thecentral conductor and an adjacent conductor of the A25sample, taken as a typical example of the ®rst family, asa function of power dissipated in the central conductor,Q, are shown in Fig. 2. A25 sample has an insulation ofFig. 1(a) type. As long as the temperature of the heatedconductor is below the He II±He I transition tempera-ture (Tk � 2:16 K), the heat transfer is in what we call``super¯uid regime'', made by conduction through theinsulating materials and by double internal convectionin the He II channels (Landau or Gorter±Mellink law).Beyond Tk, there is what we call ``mixed regime'' forwhich there are separated zones of He I and He II, He Ibeing close to the heated conductor. Measurements

Fig. 2. Typical temperature rises of conductors of a ®rst family sample

when the central conductor is heated. Sample A25: 2�KaptonÒ100 HN 50% overlap + KaptonÒ 270 LCI-1 2 mm spacing.

Fig. 1. Cable insulation schematic: (a) secondary wrapping� adhesive

tape with spacing; (b) secondary wrapping� adhesive tape alternating

with joining non-adhesive woven tape.

922 C. Meuris et al. / Cryogenics 39 (1999) 921±931

made in a He I bath at 2.2 K, reported in the same®gure, show the gain provided by the use of He II.

Section 6 shows that, in the region of very low tem-perature di�erences, DT / Q3 and the e�ect of the bathtemperature Tb on the measured temperature rise is in-versely proportional to the He II heat transfer functionin the Gorter±Mellink regime, f(T), de®ned by

q3 � ÿf �T � dT=dx;

where q is the heat ¯ux density [7,8].These two characteristics suggest that the heat is re-

moved by continuous He II paths, connecting the heli-um inside the conductor, in the machined grooves on thesurface, and the helium outside the mock-up at Tb.These channels are formed by the openings in the pri-mary wrapping at the overlapping. In fact, the insula-tion of all the mock-ups belonging to this family has asecondary wrapping with a spacing which enables theopenings of the primary wrapping on the small con-ductor faces to access the bath as shown in Fig. 1(a).These helium paths are called transverse gaps.

The e�ciency of the transfer decreases on increasingheat ¯ux transported by the He II channels, since theequivalent thermal conductivity of He II varies asf �T �=q2. Thus, in the high temperature di�erences re-gion below Tk (typically DT > 50 mK), it is observedthat the thermal behaviour deviates from a Gorter±Mellink type regime. The temperature rise in the con-ductor is the result of the combined roles of He II andthe insulating materials.

The temperature rise in the A18 sample, taken as atypical example of the second family, is shown in Fig. 3.A18 sample has an insulation of Fig. 1(b) type. Its be-haviour is di�erent from that described above by thelinear variation of the graph DT(Q). In addition, thetemperature increase of the central conductor is practi-cally identical to that when it is heated alone and when itand its neighbours are heated, while for samples of the®rst family temperature increase of the central conduc-tor is greater when the three conductors are heated, asshown in Fig. 4.

In the A18 sample case (family 2), the conductors arethermally de-coupled from each other. This character-istic suggests there are longitudinal He II conduits be-tween conductors with very small thermal resistance inthe region of low DT, and this forms a thermal shortcircuit for the heat transferred by the large faces of theconductors. In fact, the secondary insulation wrappingof the mock-ups belonging to this family is made up ofdry woven tape which enables maintenance of heliummulti-channels along the tissue threads (Fig. 1b).

In the A32 sample case (family 1), the heat removedby the large faces is transmitted to adjacent conductors.The compression of the insulation on these faces doesnot allow maintenance of helium conduits in the spacingof the secondary tape.

As shown in Fig. 4, the performance of each insula-tion depends on the total heat power to be removed andthe number of heated conductors:· At low heat power the transfer is dominated by the

small faces which are in direct contact with the bath;

Fig. 3. Typical temperature rises of conductors of a second family

sample when the central conductor is heated. Sample A18: KaptonÒ100 HN 50% overlap + impregnated ®breglass joined with dry Kev-

larÒ/®breglass.

Fig. 4. In¯uence of number of heated conductors. Tb � 1:9 K. Sample

A32 (family 1): KaptonÒ 150 HN 50% overlap + KaptonÒ 270 LCI-1

2 mm spacing. Sample A18 (family 2): KaptonÒ 100 HN 50% over-

lap + impregnated ®breglass joined with dry KevlarÒ/®breglass. Full

symbols: only central conductor heated. Open symbols: three central

conductor heated.

C. Meuris et al. / Cryogenics 39 (1999) 921±931 923

the performance of A18 is then less than that of A32since the secondary tape prevents the transverse gapsto open into the helium bath.

· At high heat power the large faces of A18 ± with a sur-face area much greater than that of the small faces ofthe conductors ± operate e�ciently due to the longitu-dinal helium conduits along the dry tissue threads.

4. Experimental results

Table 1 describes the various tested insulations. Fig. 5shows the dependence of the central conductor tem-perature rise on the heat power dissipated for di�erentinsulations made up of secondary wrapping with spac-ing (family 1), with comments on the insulation com-position. The following insulations are presented, inorder of increasing performance and excluding the in-

sulations with primary wrapping of two KaptonÒ tapeswith 50% overlap which have poor thermal perfor-mances (like A25):· Insulations with a secondary tape adhesive on both

faces (impregnated ®breglass or Kapton adhesive onthe two faces), like insulations A6, A7, A8, A9,A11, A13, A15 and A16.

· Insulations with secondary tape adhesive on the out-side face (KaptonÒ 270 or 220 LCI-1), thus non-ad-herent to the primary tapes:� primary wrapping of a KaptonÒ 200 HN (50 lm)

tape with 50% overlap, like insulations A23, A27,A30, A33 and A34 which have secondary tapeswith di�erent widths and thicknesses;

� primary wrapping of a KaptonÒ 150 HN (38 lm)tape with 50% overlap, like insulations A28, A31and A32 which have secondary tapes with di�erentwidths and thicknesses;

Table 1

Composition of tested insulations

N Primary wrappinga tape with 50% overlap Secondary wrappingb Insulation thickness

under 60 MPa (lm)

Tapes Spacing (mm)

A6 Kapton 100 HN (25 lm ´ 11 mm) Impregnated ®breglass (125 lm ´ 12 mm) 2 70

A7 Glass tissue-underlaid Kapton (70 lm ´ 12 mm) Impregnated ®breglass (125 lm ´ 12 mm) 2 93

A8c Glass tissue-underlaid Kapton (70 lm ´ 12 mm) Impregnated ®breglass (125 lm ´ 12 mm) 2 90



A12 Glass tissue-underlaid Kapton (70 lm ´ 12 mm) Dry ®breglass (80 lm ´ 12 mm) 2 125

A13c Glass tissue-underlaid Kapton (70 lm ´ 12 mm) 2 Kapton 140 XRCI-2 (2 ´ 35 lm ´ 12 mm) 2 118

A14c Glass tissue-underlaid peek (80 lm ´ 15 mm) 2 Kapton 140 XRCI-2 (2 ´ 35 lm ´ 9 mm) 2 90

A15c Kapton 100 HN (25 lm ´ 11 mm) 2 Kapton 140 XRCI-2 (2 ´ 35 lm ´ 12 mm) 2 100

A16 Kapton 120 XCI-1 (30 lm ´ 9.5 mm) Kapton 140 XRCI-2 (35 lm ´ 9.5 mm) 50% overlap 103

A18c Kapton 100 HN (25 lm ´ 11 mm) Impregnated ®breglass (125 lm ´ 12 mm)

joined with

0 109

A19c Kapton 100 HN (25 lm ´ 11 mm) dry Kevlar/®breglass (120 lm ´ m ´ 10 mm) 121

+

A20c Kapton 50 HN (12.5 lm ´ 12 mm) 2 Kapton 140 XRCI-2 (2 ´ 35 lm ´ 12 mm)

joined with

0 120

A21c Kapton 150 HN (37.5 lm ´ 11 mm) dry Kevlar/®breglass (120 lm ´ 10 mm) 105

A22c Kapton 150 HN (37.5 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 12 mm) 4 101

A23c Kapton 200 HN (50 lm ´ 36 mm)d Kapton 270 LCI-1 (67.5 lm ´ 12 mm) 4 104


A25c 2 \times Kapton 100 HN (25 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 12 mm) 2 140

A27 Kapton 200 HN (50 lm ´ 15 mm) Kapton 270 LCI-1 (67.5 lm ´ 9 mm) 2 130

A28c Kapton 150 HN (37.5 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 12 mm) 3e 118


A30 Kapton 200 HN (50 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 9 mm) 2 135

A31 Kapton 150 HN (37.5 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 12 mm) 2 107

A32 Kapton 150 HN (37.5 lm ´ 11 mm) Kapton 270 LCI-1 (67.5 lm ´ 9 mm) 2 110

A33 Kapton 200 HN (50 lm ´ 11 mm) Kapton 220 LCI-1 (55 lm ´ mm) 2 123

A34 Kapton 200 HN (50 lm ´ 11 mm)f Kapton 270 LCI-1 (67.5 lm ´ 9 mm) 2 132

A35 Kapton 200 HN (50 lm ´ 11 mm)g Kapton 270 LCI-1 (67.5 lm ´ 9 mm)h 2 130

a Wrapping in the opposite direction to groove direction simulating cable twist.b Secondary wrapping in the opposite direction to primary wrapping.c Polymerisation mould width� conductor width + 0.4 mm (0.3 mm for the other samples).d No overlapping; 2 mm wide opening on large conductor face.e Spacing occupied by ®ve 80 lm diameter KevlarÒ strands.f Two layers of joined tapes.g Two layers of joined tapes.h Same direction for secondary and primary wraps.

� ��


� insulations whose lateral faces (small faces of theconductors) were not compressed in the polymeri-sation, like insulations A22, A24 and A29 whichhave primary wrappings of a KaptonÒ 150 HNtape with 50% overlap and which have secondarytape spacings varying from 2 to 4 mm;

� insulations whose primary and secondary tapesare wound in the same direction and for whichthe position of the joints of the primary tape wascontrolled so that the transverse gaps emerge al-ways in the spacing of the secondary wrapping(A35).

The results lead to several comments. The secondarywrapping spacings do not form longitudinal conduitsbetween the conductors. This is corroborated by ob-servation of the display samples shown by the photo-graphs in Fig. 6. Fig. 6(a) and (c) present the totalinsulation ± made up by the insulation of two adjacentconductors ± in a region with a secondary wrapping foreach conductor. Fig. 6(b) and (d) are photographs in aregion where a secondary tape spacing exists for oneconductor; the total insulation is less compressed andless spread out in the conductor grooves. This contrib-utes to a local increase in the internal helium volumewithout creating longitudinal conduits between con-ductors. It is then understandable why the secondarytape spacing width and thickness are not determinantparameters.

The transverse gaps, de®ned by the openings betweenthe primary layers, are favoured by a minimum of pri-mary layers (2 layers made by 50% overlap tape forelectrical reason) and by a secondary wrapping non-adhesive to the primary wrapping. For low heat power,the heat transfer is by the transverse helium gaps on thesmall faces. For high heat power, the channels de®nedby the transverse gaps are not su�cient to remove theheat which then must cross the insulation thickness,which becomes the determining parameter.

5. Modelling of the experimental mock-ups

A thermal modelling of the experimental mock-upshas been created both to validate the role of the di�erentheat transfer paths described above and to study thesensitivity of the temperature rise to their possible di-mensions. These numerical results lead to a one-di-mensional model for the analysis of the experimentalresults.

5.1. Modelling and resolution

The main thermal paths identi®ed in the experimentalstudy are modelled by a two-dimensional geometry asshown in Fig. 7. The total thickness of the insulationunder pressure is ®xed at 110 lm per large conductor

Fig. 5. Temperature rise of central conductor when the three central conductors are heated. Secondary wrapping with spacing (family 1). Tb � 1:9 K.


face; it corresponds to the average value measured forall the tested samples. The total thickness on the lateralfaces is 130 lm per small conductor face. The heliumchannel dimensions have values which are varied in themodelling.

The thermal behaviour law for super¯uid helium inthe Gorter±Mellink regime

jqj2~q � ÿf �T � grad��!

T

was introduced in a ®nite elements code [9] in simulatingHe II by a material with heat conductivity k such that

k � f �T �j~qj2 �

f �T �grad��!

T�� 2

0B@1CA

1=3

:

The thermal conductivity values of the other materialsat 2 K are 0.011 W/m/K for the insulating material(average value of the conductivities of the di�erenttested insulations [10]), 0.11 W/m/K for stainless steel(measured value of the experimental specimen) and 0.02W/m/K for He I.

No allowance is made for the Kapitza resistance, bothat the interfaces conductor/helium and insulating ma-terial/helium. A posteriori, it has been veri®ed that thecorresponding temperature di�erences are negligibletaking account of the heat ¯ux transmitted at the solid/helium interfaces: DTmax � 3 mK for q � 8 W/m2

(Q� 50 mW, Tb� 1.9 K).The heat power dissipated in the conductors is in-

troduced as a uniform power density per unit volume inthe stainless steel. The conditions at the limits of the ®veconductors stack are:· q � 0 on the large outside faces (presence of the stain-

less steel compressive mould),· T � Tb on the small lateral faces (in contact with the

He II bath controlled at Tb).As an example, Fig. 8 shows the temperature maps of

a quarter of a sample in He I and He II, when 0.33 W/m(50 mW for a 0.15 m conductor) is deposited in each ofthe three central conductors. The shape of the isother-mals gives indications of the preferential heat paths.While in He I the heat transfer is truly two-dimensional(Fig. 8(a)), with heat exchange towards the adjacentconductors and the small faces, the He II present at theconductor surface makes the transfer practically one-dimensional (Fig. 8(b)) due to the very high equivalentthermal conductivity of He II and to the small insulationthicknesses. The heat dissipated in the heated conductoris transferred along Oy to the large faces to be evacuatedalong Ox towards the small faces by the helium presenton the large conductor faces.

5.2. In¯uence of helium distribution in the conductor andinsulation

Three con®gurations were modelled to study indi-vidually the main He II paths: con®guration a for zerolongitudinal conduit and zero transverse gap, con®gu-ration b for zero longitudinal conduit, and con®gurationc for zero internal helium and zero transverse gap. Ini-

Fig. 6. Cross-sections of display samples showing the representative-

ness of the cable insulation ((a) and (b)) by that of experimental

samples ((c) and (d)), and the ®tting of the insulation in the conductor

grooves. Type A6 insulation: KaptonÒ 100 HN 50% overlap + im-

pregnated ®breglass 2 mm spacing.

Fig. 7. 2-dimensional modelling of the main thermal paths. Con®gu-

ration a: econduit � egap � 0. Con®guration b: econduit � 0. Con®guration

c: einternal � egap � 0.


tially the overall helium volume in the stacks of con-ductors is the same for the three con®gurations andrepresents only 1% of the total material volume, which isthe estimated volume of helium in the mock-ups.

The temperature of the central conductor, taken atthe position of the temperature sensor in the experi-ments (point S2 in Fig. 8), is plotted in Fig. 9 as afunction of the heat power dissipated in the conductor.As the model is two-dimensional, the results are ob-tained by using a power per unit length of conductor. Tomake the results of calculation compatible with thoseobtained with the experimental mock-ups, the heatpower is reported for 0.15 m of conductor. The tem-perature increase of the conductor cooled by He I in apurely conductive regime is reported in the same ®gurein order to be compared to the temperature increase of

the conductor cooled by He II. Even if con®guration ahas a helium-tight insulation, its temperature increase issmaller than the conductive case due to the presence ofthe internal helium. A marked analogy can be seen be-tween experimental family 1 and con®guration b whichobeys a Q3 law for low DT (DT < 15 mK) due to thecontinuous helium path between the internal helium andthe bath. Similarly, family 2, which has a linear behav-iour due to a practically zero temperature gradient in thelongitudinal conduits is analogous to con®guration c.

The family 1-con®guration b and family 2-con®gura-tion c analogies are con®rmed by studying the in¯uenceof the number of heated conductors (compare full linesand dashed lines in Fig. 9):· Con®guration b: For low temperature di�erences

(DT < 15 mK), the heat power is completely removedby the transverse gaps after having being transferredby the internal helium to the small faces; in this DTrange, there is no in¯uence of the number of heatedconductors. For higher temperature di�erences, thetemperature rise of the central conductor is greaterwhen the three conductors are heated.

· Con®guration c: The temperature rise of the centralconductor is identical when it is heated alone or whenits neighbours are also heated. The longitudinal con-duit forms a thermal short circuit for the heat fromthe insulation of the large faces. The heat power is es-sentially removed by the longitudinal conduits afterhaving crossed part of the insulation of the largefaces.

5.3. Sensitivity of the results to the helium channeldimensions

When the heat power increases in con®guration bconductor, temperature rise deviates from the Q3 lawdue to the augmentation in the thermal resistance of theHe II channels. Part of the power is then transferred tothe adjacent conductor through the insulation of thelarge face. The same is true when the cross-section of the

Fig. 9. Comparison of di�erent con®gurations. Tb� 1.9 K. Con®gu-

ration a: einternal � 10 lm. Con®guration b: einternal � 10 lm,

egap � 1:5 lm. Con®guration c: econduit � 10 lm. Solid lines and full

symbols: only central conductor heated. Dashed lines and open sym-

bols: three central conductors heated.

Fig. 8. Temperature maps when the three central conductors are heated (Q� 50 mW in each). Con®guration b: egap � 1:25 lm, einternal � 10 lm,

econduit � 0. (a) He I: Tb � 2:2 K, DTmax � 2:4 K, DTS2� 2:3 K. (b) He II: Tb � 1:9 K, DTmax � 68 mK, DTS2

� 13 mK.


transverse gaps decreases as shown in Fig. 10 (solidcurves). It is seen that the temperature rise is highlysensitive to the transverse gap dimensions which can beunderstood since DT, at the ends of a He II channel withcross-section A, varies with A3. On the other hand, thetemperature rise of the conductor is not very sensitive tothe cross-section decrease of the internal helium as longas the transverse gap thermal resistance is larger thanthe one of the internal helium (broken curves).

Temperature rise of the con®guration c conductordeviates from the linear equation when either the heatpower increases or the longitudinal conduit cross-sec-tion decreases since the thermal resistance of the longi-tudinal conduit is no longer zero (Fig. 11).

From the numerical calculations it appears that, forall the studied channel dimensions, 92% of the heatpower dissipated in the conductor is passing by the largeface for con®guration b and 94% for con®guration c.The remaining 8% and 6% power are passing directlythrough the small faces.

6. Characterisation of the experimental heat transferpaths

The numerical calculations have shown that the heattransfer is one-dimensional for insulated conductors inHe II. The experimental con®guration can then be rep-resented by a network of equivalent thermal resistances,shown in Fig. 12. The experimental results (DT asfunction of Q curves) are analysed to determine theseequivalent resistances values and thus the dimensions ofthe di�erent helium paths in the conductor and the in-sulation.

6.1. Secondary wrapping with spacing (experimentalfamily 1)

The solving of the equivalent thermal circuit leads, inthe low temperature rise region where f(T) can be con-sidered as constant and conduction in solid negligible, tothe equations:

DTS2� G

f �Tb�Q3; �1�

G � Ggap � Ginternal

� lgap

2egap � Lÿ �3

� 0:923`

4� 2einternal � L� �3 ; �2�

where Q is the total heat power dissipated in L � 0:15 mof conductor (length of mock-ups) 92% of which is re-moved by the two large faces of the conductor, and G isa parameter depending on the geometry as length/(cross-section)3, characterising all the helium channels con-necting the internal helium and the external bath. These

Fig. 10. In¯uence of the transverse gap dimension and internal helium

volume of con®guration b; only central conductor heated; Tb � 1:9 K.

Fig. 11. In¯uence of the longitudinal conduit dimension of con®gu-

ration c; only central conductor heated; Tb � 1:9 K.

Fig. 12. Network of equivalent thermal resistances.


channels consist of transverse gaps with equivalentlength and cross-section that are, respectively, lgap andegap � L, in series with internal helium cavities withequivalent cross-section einternal � L, uniformly heatedover the length ` � 17 mm (dimension of the large faceof a conductor). Factor 4 comes from the uniformheating of the internal helium.

Remark on equivalent dimensions: All the real trans-verse gaps, with lengths lgap;i and cross-section Sgap;i, actin parallel for the heat transport [11]. The equivalentlength lgap and equivalent cross-section, egap � L, aresuch that

egap � L

l1=3gap

�X

i

Sgap;i

l1=3gap;i

:

The values of the various quantities derived from theexperimental results are equivalents calculated for thelength of the mock-up. The equivalents per unit lengthof conductor would be G0 �mÿ2� � G� L3 and egap (gapcross-section per unit length of conductor).

The low DT region is de®ned as that for which DT isproportional to Q3, a law which shows the presence of acontinuous helium path between the conductor and thebath (see Fig. 13). For each bath temperature, the ex-perimental curves are approached by a curve DT � CQ3.The validity of the ®t is in the interval around [2, 20mK]. For DT < 2 mK, the mutual friction regime(Gorter±Mellink) cannot be fully developed or is theKapitza resistance, between the conductor and the in-ternal helium, non-negligible.

An example is shown in Fig. 14 and in Table 2. For agiven sample, the invariability of the coe�cient G with

the bath temperature validates the result. The curves forwhich the value of the exponent n, giving the best ®t inthe approximation DT � C0Qn, is above 3.1 or below 2.9are systematically rejected. The coe�cient G is calcu-lated from the average of the values at the di�erent se-lected bath temperatures. For the given example,GA32 � 1:0� 0:2� 1016/m5.

A high G coe�cient characterises a relatively poorperformance insulation (low permeability). This coef-®cient is a very sensitive parameter since it varies by afactor of 400 between the least permeable sample�GA25 � 2:5� 0:4� 1017/m5) and the most permeablesample �GA35 � 6:0� 2:0� 1014/m5). The uncertaintyis essentially due to the scattering of the results withthe bath temperature. To this uncertainty the manu-facturing reproducibility due to the precise ®tting ofthe di�erent materials, an order of 50%, must beadded.

Realistic orders of magnitude for the internal heliumvolume dimensions can be obtained by calculating theequivalents in terms of thermal resistance in the Gorter±Mellink regime. Taking into account the dimensions ofthe machined grooves on the conductor surface, thevalue of the equivalent thickness of the internal helium iseinternal � 2 lm. This is a maximum value calculatedwithout taking account of the penetration of the insu-lation in the grooves. From this value, we can calculate aminimum value of Ginternal and derive a minimum valueof the equivalent thickness of the transverse gap egap byEq. (2). The resulting values vary between 0.27 and1.7 lm (egap � 0:81 lm for sample A32 given in theexample).

Fig. 13. Identi®cation of so-called ``low DT '' study interval (solid line);

sample A32; Tb � 1:9 K.

Fig. 14. Approximation of curves in low DT region at di�erent bath

temperatures; sample A32; three heated conductors.


6.2. Secondary wrapping with dry woven tape (experi-mental family 2)

Solving the circuit equivalent to con®guration c leadsto the following equation, in the low heat power regionfor which the thermal resistance of the longitudinalconduits is negligible:

DTS2� 0:94einsulator 1

2L`kinsulator 1

Q; �3�

where einsulator 1 is the average thickness of insulationwith conductivity kinsulator 1, which the heat must crossbefore reaching the longitudinal helium conduits in theinsulation of the large face (6% of the heat power isremoved by the small faces).

The equivalent transfer coe�cient of the large facesinsulation, �k=e�insulator 1, can be determined from theapproximation of the experimental curves by using Eq.(3). For each sample, the low DT region is de®ned asthat for which DT is proportional to Q, a law whichshows a transfer across the solid insulating material anda longitudinal He II conduit with zero thermal resis-tance. For the Fig. 3 example (sample A18), the intervalis around [5, 20 mK]. For DT > 20 mK, the thermalresistance of the longitudinal helium conduit is nolonger negligible and the slope of the curve DT(Q) in-creases. For this sample, the coe�cient �k=e�insulator 1 is114� 9 W/m2/K. This is an overall experimental trans-fer coe�cient taking account of the e�ective exchangesurface area.

7. Conclusions

The analysis of the experimental results with respectto the variation of DT as a function of Q, the e�ects ofthe bath temperature and the number of heated con-ductors, has made it possible to determine the dominantheat transfer modes through the insulation (insulatingmaterial ± He II composite). In insulations with a sec-ondary wrapping with spacings, the heat is removed, atlow heat power, by the He II gaps de®ned by theopenings in the primary wrapping overlaps. These gapswhich provide a permeability at right angles to the in-

sulation of the conductor small faces, are favoured by aminimum number of primary layers (two) and by asecondary wrapping non-adhesive to the primarywrapping (KaptonÒ type LCI-1). For high heat power,the thermal resistance of the He II gaps increases andthe insulator takes over the heat transfer. An increasein the primary tape thickness thus reduces the overalle�ciency. Insulations consisting of a secondary wrap-ping that is made up of a non-adhesive woven tape haveHe II conduits along the tissue threads. These multi-conduits provide a permeability longitudinal to theinsulation of the large conductor faces. The heat is es-sentially removed by these longitudinal conduits afterhaving crossed the thickness of the primary tapes.

Numerical modelling made possible the identi®cationof the role played by the di�erent possible locations ofhelium in the conductor and insulation, the sensitivitycalculation of the results to variations of these heliumcavities and the insulation thickness, and to obtainqualitative consistency with the experimental results: theresults clearly show the two families with their respectivecharacteristics.

The processing of the experimental results, based on asimpli®ed thermal model suggested by the numericalmodelling, made it possible to quantify the thermalpermeability to He II of the di�erent tested insulationassemblies in terms of equivalent thermal quantities (Gand k=einsulator 1). The results have governed the choice ofthe cable insulation of the LHC magnets: 50 lm thickprimary tape with 50% overlap and non-¯owing sec-ondary tape, adhesive on outside face, with a 2 mmspacing. The thermal behaviour of this insulation is likeinsulation A30. Fig. 5 shows that it remains somemargin for a future higher performance machine.

Acknowledgements

The authors thank R. Aymar, Director of the ``Di-rection des Sciences de la Mati�ere'' of CEA, who en-couraged this research programme, L. Burnod fromCERN for fruitful discussions, R. Gaubert and A.M.Puech from CEA for the sample preparations andmeasurements, C. Pes for the development of the He II

Table 2

Approximation of curves at di�erent bath temperatures in the low DT region. Sample A32

Tb (K) Fit interval (mK) Approximations f(Tb)a (W3/m5/K) G � Cf �Tb� (mÿ5)

DT � C0Qn; exponent n DT � CQ3; C (K/W3)

1.7 �4; 8� 2.54 ± ± ±

1.8 �2; 4� 2.90 1040 1:17� 1013 1:2� 1016

1.9 �3; 10� 3.00 668 1:65� 1013 1:1� 1016

2.0 �6; 13� 2.99 712 1:5� 1013 1:0� 1016

2.05 �6; 14� 3.10 963 0:89� 1013 0:86� 1016

2.1 �5; 16� 3.32 ± ± ±

a Average value of the helium heat transfer function between Tb and the ®t interval.


numerical model in the CASTEM code, and the uni-versity trainees S. Fuzier, S. Mamou and B. Molinazzifor their e�cient participation in the numerical calcu-lations and processing of the experimental results. B.B.expresses his thanks to Jeumont Schneider for its sup-port in his doctorate thesis work.

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Heat transfer in electrical insulation of LHC cables cooled with superfluid helium