Cooling of oil-filled cable joints usingheat pipes

J.P. Ruiter and A.W. Thus

Indexing term: Cables and overhead lines

Abstract: The paper describes a method to decrease the maximum temperature of a high-voltage cable joint bytransporting heat from the joint to the adjacent cable, using heat pipes. The calculated decrease of the tem-perature peak in the joint is about 30%. This corresponds, however, with an increase in the permissible currentrating of only 5%, which is in good agreement with the measured values. The experimental results wereobtained using a simplified thermal model of a joint. Further increase in the current rating may be obtained byenlarging the diameter of the oil channel in the hollow conductor.

1 Introduction

All high-voltage cable circuits above a certain lengthrequires joints. In general, these joints have a thickerdielectric than the adjacent cable. Particularly with forcedcooling, when the thermal limitation of the soil is substan-tially reduced and, consequently, the maximum permissibleconductor temperature plays a dominating role, additionalcooling of the joint may become necessary to maintain theconductor temperature at an acceptably low level.

Some methods of accessory cooling are already avail-able and used in practice. The application of thesemethods will be dependent on the method of forcedcooling and type of cable. For a conductor-cooled cablesystem, special cooling arrangements at joints are usuallynot necessary, but, for externally or surface-cooled systems,in general some additional cooling will be required. Forexternal cooled cables the joint may be cooled in a similarway as the cable, using cooling water in pipes near thesurroundings of the joint. For oil-filled cables, the tem-perature in the joint can be reduced by axial movement ofthe oil within the central oil duct. A more novel method,also based on the principle of 'thermal smoothing', but farless complicated, is the inclusion of heat pipes within theconductor in the joint. The application of heat pipes forcooling sealing ends has been rather successful, as indi-cated in Reference 1.

2 Theoretical model

2.1 GeneralA heat pipe is a device with which heat can be transportedat small temperature differences. The heat pipe consists ofa closed pipe, containing a capillary structure and aworking fluid. Heat is transported by the working fluid,which is evaporated at the heated side and condensed atthe cooled side of the heat pipe. The condensate isreturned to the evaporator by the capillary structure. Incase of vertical heat pipes, for instance in a sealing end, thecondensate is returned by gravity.

To calculate the effect of the heat pipes, the temperaturedistribution in the joint and in the cable must be known. Anormal high-voltage joint is, as is well known, a rathercomplicated construction. To achieve the required dielec-tric integrity, crepe paper tapes are applied adjacent to theconductor connection and preshaped profiled paper rolls

Paper 3484C (P8), first received 6th March and in revised form 21st August 1984The authors are with KEMA Laboratories, PO Box 9035, 6800 ET Arnhem, TheNetherlands

are applied thereafter. For the purpose of this study, calcu-lations need not necessarily be carried out for such a com-plicated model, but could be based on a simplified modelas shown in Fig. 1 (see Appendixes 9.1-9.6). According tothis model, pr is the radial heat flux per unit axial lengthfrom the conductor to a coaxial cylindrical isothermal areawith a constant soil temperature 7̂ and a radius rs (Fig. 2).

I I

Tp- EQp

prdx = k(T-Ts)dx-Wdx

pdx "•Or

ppdx = kp(T-Tp)dx

Fig. 1 Cable joint modela Simplified model of cable and joint with surrounding soilb Heat balance of conductor element

I_!Li

Ps=Pr*W

W

Fig. 2 Model for calculating the thermal conductance between conduc-tor and soil

340 IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984

The radial thermal conductance per unit axial length of thejoint kj or the cable kc is defined as the total radial conduc-tance per unit axial length between conductor and soil at adistance of rs. In accordance with IEC publication 287 [2],it is assumed that the dielectric losses W are generatedhalfway the dielectric. The radial heat flux per unit axiallength from the conductor to the heat pipes is pp. Thethermal conductance kp refers to the conductance betweenthe conductor and the interior of the heat pipes, per unitaxial length.

Axial heat transport from the joint to the cable isassumed to occur only by conductance through the con-ductor and by the heat pipes.

As an example, in the following Subsection, the increaseof the permissible current rating is calculated for heatpipes in a joint in a 400 kV oil-filled cable with the par-ameter values given in Table 1 (column a).

2.2 Joint without heat pipesThe permissible heat production in the conductor isobtained from the condition that the conductor tem-perature has a maximum value at the centre of the joint.

According to the aforementioned model (Section 2.1and Fig. 2), the total radial thermal conductance betweenthe conductor and the soil, at a distance of rs, is

k =T -T.

(1)

Hence, the maximum permissible heat flux from the con-ductor to the surrounding soil is

Prj = HTmax - Ts) -Wj = 0.93 x (85 - 20) - 13

= 47.5 W/m (2)

where the index; refers to the joint centre.

Table 1 : Cable parameters

Radial thermal conductance between conductor and soilat a distance of rs (or sheath*) at the jointRadial thermal conductance between conductor and soilat a distance of rs (or sheath*) at the cableDielectric losses in jointDielectric losses in cableElectric resistance of conductorCross-sectional area of conductorThermal conductivity of copper conductorEffective length of half of the jointSoil temperatureMaximum allowable conductor temperature in joint centreHeat-pipe lengthThermal conductance between conductor and heat-pipeinterior

*/

kc

w iWc

RSA/,T,Tmgx

lpma'

kD

W/mK

W/mK

W/mW/mQ/mm2

W/mKm°C°CmW/mK

0.93

1.37

1312

1.257.1O"5

2.10-3

3930.75

2085

2.253.0

0.91

1.61

001.257.102.10-3

3930.75

3587

2.253 0

* Column b refers to the same cable as column a. However, the cable in the experimental setup (columnb) was operated at low voltage, so there were no dielectric losses. Furthermore, this cable was notinstalled in the soil, but in air. The conductance between conductor and sheath is derived from eqn. 18,where W=0and R = 0.

Table 2: Thermal resistances

Joint Cable

Ratio of radius of isothermal cylinder and sheath radiusThermal conductivity of soilRadial thermal resistance over soilRadial thermal resistance over sheathRadial thermal resistance over insulationDielectric lossesRadial thermal conductance between conductor andsoil, at a distance of r.

—W/mKmK/WmK/WmK/WW/mW/mK

'J',h = 3

^ = 1 - 8 2Rs = 0.1Q

Rsh = 0.00/?, = 1.10Wt=13k, ~ 0.93

rjrsh = 6As = 1.82fls = 0.15

/?,l, = 0.12R, = 0.50

Wc = 12kc =1.37

Table 3: Integration constants

Region 1

joint

0 ^ x ^ /,.* , = k, = 0.93H/ = n/. = 13

kp 0.0

a, 1.088A, -6.070S, -6.070C, 97.140

3.0

2.236-0.671-0.67186.341

Region 2

heatjoint

Ij <>

kP

a2

A2

B2

c2

pipe outside

k^="i .3',-wc = i:

0.0

1.3200.000

24.48371.635

lI

3.0

2.358-0.00718.98979.433

Region 3

remote cable

1kP

* 3A3

B3

c3

= WC = 12

0.0

1.3200.000

24.48371.635

•

3.0*

1.3200.000

55.31075.197

W/mKW/m

W/mK

m-1

KK°C

* In the remote cable (region 3) there is no heat pipe, so kp3 = 0; the con-stants B3 and C,, however, are functions of the permissible heat production pand, hence, also functions of ko

IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984 341

Furthermore, heat is transported from the joint to theremote cable by conduction through the copper conductor.At the end of the joint (x = 0.75 m) this heat flux amountsto Qa = 9.4 W (see Fig. 3b and eqn. 29).

According to eqn. 30, the gradient of the axial heat fluxby conduction at the joint centre (x = 0) is pa = 11.3 W/m.Hence, the total permissible heat production in the jointcentre is

Pjm = Prj + Pa = 47.5 + 11.3 = 58.8 W/m (3)

The maximum permissible current rating / in the joint isderived from

Pjm = I2R

According to eqns. 2, 3 and 4,

- Ts) -WJ + Pa]1/2

(4)

(5)

and amounts to / = 2162 A.The radial thermal resistance in the cable is smaller

than in the joint. The maximum permissible heat flux fromthe cable to the soil is

Prc = K(Tmax - Ts) -Wc = 1.37 x (85 - 20) - 12

= 77.1 W/m (6)

The permissible current rating Ic in the cable follows from

Prc = IcR (7)

and amounts to Ic = 2476 A. Therefore, the permissiblecurrent rating in the cable is 15% larger than in the joint.

Fig. 3 Conductor temperature T and axial heat flux Qa, by conductionfrom joint to cable, as a function of the axial distance x from the jointcentre, when no heat pipes are appliedkp = 0 W/mK, kj = 0.93 W/mK, kc = 1.37 W/mK, W} = 13 W/m, Wc = 12 W/m,T, = 20°C, 7 ^ = 85°C

a Conductor temperatureb Axial heat flux through conductor

342

The temperature difference between the conductor inthe joint and in the remote cable amounts to (see Fig. 3aand eqn. 28)

AT = 8 5 - 7 1 . 6 = 13.4 K (8)

2.3 Joint with heat pipesIf the joint is equipped with a heat pipe according to theassumptions mentioned in Table 1, column a, the tem-perature in the heat pipe amounts to Tp = 81.4°C (see eqn.39). According to eqn. 38, this pipe will transport a heatflux of Qp = 6.5 W from the joint to the cable. The heattransport from the conductor in the joint centre to the heatpipe follows from Fig. 1:

PP = kP(Tmax - Tp) = 3.0 x (85 -81.4)

= 10.8 W/m (9)

As a consequence of the temperature smoothing, however,the heat transport by conduction from the joint to thecable decreases from Qa = 9.4 W to Q* = 6.1 W. This cor-responds to a heat flux gradient of p* = 5.3 W/m in thejoint centre. Assuming the same maximum conductor tem-perature as in the joint without heat pipes, the permissibleheat production in the joint is now

Pfm = Prj + Pt + PP = 47.5 + 5.3 + 10.8

= 63.6 W/m

The permissible current rating /* follows from

(10)

P% = (11)

From eqns. 2, 9, 10 and 11, it can be derived that

I* = illkj(Tmax-Ts)- kp(Tmax -

(12)

or /* = 2250 A.The temperature difference between the conductor in

the joint and in the remote cable is now AT = 9.8 K (seeFig. 5 for kp = 3.0 and eqn. 28).

Hence, by application of these heat pipes, the maximumpermissible current rating in the joint is increased by only4.1% from 2162 A to 2250 A. The temperature differencebetween the conductor in the joint and the conductor inthe remote cable is decreased by 27% from 13.4 to 9.8 K.

3 Effect of varying parameters

The increase of the permissible current rating stronglydepends on the thermal conductance between the conduc-tor and the interior of the heat pipe. This thermal resist-ance is mainly caused by the oil film in the hollowconductor. The thermal conductance kp, as calculated inSection 9.5, is presented in Fig. 4.

From Fig. 4, it appears that the oil film has to be as thinas possible. In joints, however, the radial dimensions of theoil channel should be such that, when the temperaturechanges, the oil flow should be possible without too muchpressure drop.

In larger oil channels, larger heat pipes can be applied.If the oil-film thickness is kept constant, the thermal con-ductance kp will increase as shown in Fig. 4.

In Fig. 5, the gain in current rating A/// and the tem-perature difference between the conductor in the joint andin the remote cable AT is presented as a function of theconductance kp. In this Figure the heat flux Qp transportedby the heat pipe is also shown. A value of kp = 10 W/mK

IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984

Fig. 4 Effect of oil film thickness 5 on thermal conductance kp betweenconductor and heat pipedp = outside diameter of heat pipe, mm

15

10

\ AI/I

/ / "CT Q p

, . — — — •

• •

AT

- L

20-

.

5 10-Q.

O

10

5 i

10 20 30ko. W/mK

Fig. 5 Increase of permissible current rating AI/I, decrease of tem-perature difference AT between conductor in the joint and conductor in theremote cable, and heat flux Qp through the heat pipe, as a function ofthermal conductance kp between conductor and heat pipe interior

lp = 2.25 m, kj = 0.93 W/mK, kc = 1.37 W/mK, W, = 13 W/m, Wc = 12 W/m, T =20°C, Tmx = 85°C

Fig. 6 Increase of permissible current rating AI/I as a function of theheat-pipe length lp

0.93 W/mK, kc = 1.37 W/mK, W} = 13 W/m, Wc = 12 W/m, T, = 20°C,j

85CC

seems a reasonable aim. Higher values only give a slightimprovement.

In Fig. 6 the increase in permissible current rating isshown as a function of the heat pipe length lp. It appearsfrom this Figure that the heat pipes should have a lengthof about 2 to 3 m.

If heat pipes are applied with a length of 2.25 m, theconductor temperature difference and axial heat fluxthrough the conductor will decrease as shown in Figs. 7and 8.

As can be seen in Fig. 8, the axial heat flux Qa by con-duction will decrease, as the temperature profile issmoothed by heat pipes.

The permissible current rating / and the differencebetween conductor temperature in the joint and in thecable AT are also affected by the soil temperature Ts, ascan be seen in Fig. 9. However, within the limits of 0 <Ts < 50°C, the gain in permissible current rating AI/I ishardly influenced by the soil temperature.

85

kp=1000W/mKkp=25

kD=5

kp=1

kD=0

Fig. 7 Conductor temperature T as a function of the axial distance xfrom the joint centre, when heat pipes are applied

lp = 2.25 m, kj = 0.93 W/mK, kc = 1.37 W/mK, W} = 13 W/m, Wc = 12 W/m, Ts =20°C, T^ = 85°C

Fig. 8 Axial heat flux Qa through the conductor, as a function of theaxial distance xfrom the joint centre, when heat pipes are appliedlp = 2.25 m, kj = 0.93 W/mK, kc = 1.37 W/mK, W} = 13 W/m, Wc = 12 W/m, T, =20°C, Xn., = 85°C

3000

2600

2200

1800

50

Fig. 9 Effect of soil temperature Ts on permissible current rating I, onincrease in current rating AI/I and on temperature difference AT betweenconductor in the joint and conductor in the remote cablelp = 2.25 m, fc; = 0.93 W/mK, *c = 1.37 W/mK, Wj = 13 W/m, Wc = 12 W/m,T«« = 85°C, k = 3.0 W/mK

IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984 343

4 Experimental set-up

The effect of heat pipes was measured in an experimentalsetup as shown in Figs. 10, 11 and 12. Before explainingthe experimental installation in detail, it must be statedthat the design of the experimental joint was rather simpli-

coolingwater

220V-

water-cooled hollow copper duct

T18) heat pipe (T19) (T2o) joint

Fig. 10 Scheme of the experimental setupTo = ambient temperatureT,—> T16 = conductor temperaturesT,7—> T22 = heat-pipe surface temperatures

6.00m

T23 —* T26 = cable surface temperaturesT21 = copper-duct temperature

= cooling-air temperature= current rating

thermal isolation

papercopper conductorcopper pipeoil channel

l&&<&<&<^^7copper connecting tube

1.5m

Fig. 11 Simulated joint in the experimental setup {detail of Fig. 10)

Fig. 12 Experimental setup

fled. The main purpose was to simulate the essentialthermal function of the joint, and without further com-plications being thermally representative for the practicaljoint. For this purpose, two 400 kV oil-filled cable pieces(see Table 1, column b), with a length of 3 m each, werejoined together. The coupling between the conductors wasmade in accordance with real jointing practice. Thestainless-steel helix was removed from the oil channel andreplaced by a copper pipe (diameter 15/13 mm). While

making a joint, this copper pipe is used to prevent solderflowing into the oil channel. In real joints this pipe isabout 1 m long; in this setup the copper pipe was as longas the cable (6 m). Between the thermal isolation and thepaper insulation a special tape is applied to avoid oilleakage.

The construction material of the joint was simulated bythermal isolation (see Fig. 11) in such a way that the con-ductor temperature in the centre of the joint was about90°C at a current rating of 2100 A (see Fig. 15).

The heat pipes used were 2.25 m long and consisted ofcopper pipes and a stainless-steel wick (see Table 4). Waterwas used as a working fluid. The outer diameter of theheat pipe was 9.5 mm, so the oil-film thickness was 1.75mm. To improve the thermal conductance, a copper wirewas coiled around the heat pipe. Furthermore, the thermo-couple wires were wrapped between this copper helix (seeFig. 13). The thermal conductance between the conductorand the heat-pipe interior, per unit axial length, is obtainedfrom the simplified scheme of Fig. 14 and amounts to kp =3.0 W/mK (see Appendix 9.6).

The experiments were performed at low voltage (6 V),and so there were no dielectric losses. The electrical resist-ance was mainly caused by the clamps. The dissipationheat (13 kW) was removed by cooling water through thehollow copper duct (see Fig. 10). The left cable end is alsopartly cooled by this cooling water. Hence, for reasons ofsymmetry, at the right cable end an identical situation isaimed at, by applying a water-cooled dummy clamp.

344 IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984

Table 4: Heat pipe parameters

Outside radius ro mInside radius r, mWall thickness (copper) tp mWick thickness (stainless steel) tw mVapour core radius rv mEvaporator length Le mCondenser length Lc mScreen mesh number N m - 1

Screen wire diameter d mWick crimping factor s —Vapour temperature TY K

Drag coefficient /„ Re, —Vapour frictional coefficient Fv s/m4

Wick porosity e —Effective thermal conductivity of wick ke W/mK

Vapourisation heat of water r J/kgDensity of water vapour pv kg/m3

Dynamic viscosity of water vapour //„ kg/msThermal conductivity of water A, W/mKThermal conductivity of copper wall Ac W/mKThermal conductivity of stainless steel wick Aw W/mK

Thermal conductivity of cable oil Aol W/mKThermal conductivity of thermocouple wire Atc W/mKThermal conductivity of oil-impregnated Aop W/mKpaper

The current rating was kept constant within 1 % by twotransformers, which were cooled by air.

paper*oilcopper pipe

Ao p=0.UXc =393XOI =0.14

160.160.611.41

z.3 * 100.2911 .2x i o - 6

0.6739315

0.140.300.14

Fig. 13 Heat pipe, wrapped with copper helix and thermocouple wires

— oil filmheat pipe

thermocouple wire Ate =0.30

copper wire(<}»1.2) Ac =393

Fig. 14 Model for calculation of thermal conductance kp between con-ductor and heat pipe in experimental setup

The heat pipe is wrapped with copper wire and thermocouple wires (all dimensionsin mm; thermal conductivity / in W/mK)

5 Results

The measured temperature profiles of the conductor and ofthe heat-pipe surface, at a current rating of 2100 A, arepresented in Fig. 15.

The temperature profiles (especially for the measure-ments with heat pipes) are not symmetrical. Probably thisis caused by imperfections in the setup, such as unequalboundary effects.

As can be seen from eqns. 5 and 12, there is a linearrelationship between the square of the current rating I2

and the difference between the maximum conductor tem-perature Tmax and the soil temperature 7^. As in this casethe cable was installed in air, the soil temperature isreplaced by the ambient temperature To. In Fig. 16 thisrelationship is shown. From these measurements, it can bededuced that, by the application of these heat pipes, thecurrent rating can be increased by 4%, while the tem-perature difference (Tmax — To) is kept constant.

As the experimental setup was only 6 m long, the tem-perature difference between the joint centre and the remotecable could not be measured. Therefore, in Fig. 17, thetemperature difference between the conductor in the jointcentre and the conductor at the end of the heat pipe ispresented. At a current rating of 2100 A, the temperaturedifference by application of the heat pipes decreases by36%, from 18.2 K to 11.7 K.

90 -

-2 -1 0

distance from joint centre, m

Fig. 15 Measured temperature profiles of conductor and of heat-pipe surface, at a current rating of 2100 A

• without heat pipes D heat-pipe surface x with heat pipes

IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984 345

501

I2,1O5A242 44 46

Fig. 16 Measured difference between maximum conductor temperatureand ambient temperature against square of current rating

20without heat pipes

1900 20001 A

2100

Fig. 17 Measured difference between conductor temperature in jointcentre and at end of heat pipe ATp against current rating I

6 Comparison between theory and experiment

The increase of the permissible current rating and thedecrease of the aforementioned temperature difference hasbeen calculated for the experimental setup.

As the cable in this setup was not installed in the soil,but in air, the soil temperature 7̂ in the calculations isreplaced by the cable surface temperature Tcs, which isassumed to be constant. It appears from Fig. 18 that, for a

15

0 10 20 30kp,W/mK

Fig. 18 Calculated increase of permissible current rating A/// and tem-perature difference ATp between the conductor in the joint and the conduc-tor at the end of the heat pipe, in the experimental setup

lp = 2.25 m, kj = 0.91 W/mK, kc = 1.61 W/mK, W, = 0 W/m, Wc = 0 W/m, T =35°C, Tmax = 87°C

thermal conductance of kp = 3.0 W/mK, the calculatedincrease in the current rating (A/// = 4.9%) agrees withthe measured value (4%). Also, the decrease of the differ-ence between the conductor temperature in the joint centreand at the end of the heat pipe (calculated:13.4-8.0 = 5.4 K, or 40%, and measured:18.2 — 11.7 = 6.5 K, or 36%) is in good agreement. Thecalculated and measured absolute values of the tem-perature difference show some disagreement. Probably thisis caused by boundary effects of the relatively short labor-atory test arrangement.

The calculated heat transport by the heat pipe, from thejoint to the cable, amounts to 7.6 W.

These results are in good agreement with experimentalresults of Pirelli General [4]. Crockett and Yates mea-sured, in their rig, a temperature difference between jointand remote cable of 108 — 85 = 23 K. Application of heatpipes with a diameter of 19 mm reduced this difference to96 - 85 = 11 K; hence a reduction of 48%.

7 Conclusion

The application of conventional heat pipes in cable jointsreduces the difference between the maximum conductortemperature and the remote cable temperature by about30%; however, this only results in about 5% higher per-missible current rating, assuming the maximum conductortemperature of 85°C not to be exceeded. Therefore, for thistype of heat pipe there is only a marginal effect; applica-tion in joints does not seem to be very useful.

Larger improvements could be obtained by decreasingthe thermal resistance between the conductor and the heatpipes.

Referring to the requirement that the oil channel shouldbe free for oil transport, it could be useful to investigatethe possibilities of an annular heat pipe that fits closelyinto the hollow conductor, and lets the oil flow throughthe central channel.

Better results can also be obtained by increasing the oilchannel diameter. However, for the cable and joint asdescribed in Table 1, column a, the maximum improve-ment in permissible current rating that could be achievedis limited to 15%.

8 References

1 RUITER, J.P., and v. HULST, L.P.D.M.: Toepassing van warm-tepijpen bij de koeling van 400 kV-eindsluitingen', Elektrotechnniek,1979, 57, (11), pp. 777-781

2 'Calculation of the continuous current rating of cables (100% loadfactor)' (IEC Publication 287, 1969)

3 CHI, S.W.: 'Heat pipe theory and practice', (McGraw Hill, New York,1979), Chap. 2

4 CROCKETT, A.E., and YATES, G.: 'Cooling of accessories of highvoltage cable systems'. Proceedings of the 2nd international conferenceon progress in cables and overhead lines for 220 kV and above, Sep-tember 1979, pp. 256-261

9 Appendixes

9.1 Thermal conductance between conductor and soilAccording to the model described in Section 2.1, the totalradial thermal resistance between the conductor and thesoil at a distance of rs, per unit axial length, is defined as

T -Ts A7\ + AT AT(13)

(The notation refers to the Figs. 1 and 2; see also Tables 1and 2.)

346 1EE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984

With

AT2 = \Rt{pr + W)

AT3 = Rsh(pr + W)

AT4 = Rs(pr + W)

it follows that the radial thermal conductance between theconductor and the soil at a distance of rs, per unit axiallength, is

(14)

(15)

(16)

(17)

1 +- Ts)

Rt R{ + Rs Rs

(18)

where the radial resistance over the soil is

Ink.(19)

As k is only a weak function of T, k can be considered asconstant: kj in the joint and kc in the cable.

9.2 Local temperature and axial heat flux in theconductor

In case of thermal equilibrium, the heat balance of a con-ductor element dx follows from Fig. \b:

Qa + pdx = Qa —r1 dx + prdx + pp dx (20)

whereQa = axial heat flux through conductor

p = heat production in conductor per unit axial length

According to eqns. 13 and 18, the radial heat flux from theconductor to the insulation is

pr = k(T -Ts)-W (21)

The radial heat flux from the conductor to the heat pipe is

pp = kp(T - Tp) (22)

whereT = local conductor temperatureTp = temperature inside the heat pipe

The axial heat flux through a conductor with a cross-section S and a thermal conductivity k is

and the gradient of the heat flux

dx " " X & dx2

From eqns. 20, 21, 22 and 24 it follows that

d2T

dx2

where

- a2T + a2C = 0

" ~ kS

and

fi s ' p p ' r '

IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984

(23)

(24)

(25)

(26)

(27)

Solution of this differential equation gives the local con-ductor temperature

T = A exp (ax)+ B exp ( -ax) + C (28)

the axial heat flux by conduction

Qa = -kSa[A exp (ax) - B exp ( - ax ) ] (29)

and the gradient of the heat flux

dx= pa = — kSa\A exp (ax) -I- B exp ( — ax)] (30)

The calculations are made for three regions as indicated inFig. la (see Table 3). The integration constants areobtained from the following boundary conditions:

x = 0 ^ Qal = 0

x = lj^ 71 = T2

(31)

(32)

(33)

* = lp^Qa2=Qa3 (34)

x = /p-> T2 = T3 (35)

x = co^Qa2=0 (36)

The combination of eqns. 29 with 31 and 36 results in

Ax = Bx and A3 = 0 (37)

The other constants are given in Table 3 for the values asmentioned in Table 1, column a.

9.3 Heat-pipe temperatureThe temperature Tp at the interior of the heat pipe isobtained from the heat balance of the heat pipe. The heatflux transported by the heat pipe is

QP = | M*i " Tp)dx = WP ~ T2)dx

J'J(38)

The heat-pipe temperature is derived from the com-bination of eqn. 28 and 38:

Tp = — j - [exp (uylj) - exp (-a1l)'] +

— - [exp (a2 / ) - exp (a2 / •)] +

B2—— [exp ( - a 2 I) - exp ( - a 2 /p)] +«2'p

CJj + C2(lp - lj)L

(39)

9.4 Permissible current ratingThe permissible current rating is related to the permissibleheat production in the conductor, which is limited by themaximum permissible conductor temperature in the centreof the joint (x = 0). According to eqns. 28 and 37:

y — Tm (40)

Substituting Cx according to eqn. 27 where k = kj andW = Wj gives the maximum heat production in the con-ductor at the joint centre:

Vim = (Tmax - 2At)(kj + kp) - (kj Ts kpTp+ (41)

347

The permissible current rating is

(42)

9.5 Thermal conductance between conductor and heatpipe

The heat flux from the conductor in the joint through theheat pipe to the conductor in the cable meets with thethermal resistances over the oil film (Rol) and over the heatpipe (Rhp). So the total thermal resistance for this heat fluxis

R = 2RP = 2Rol + Rh (43)

where Rpj(pc) is the thermal radial resistance between theconductor and the interior of the heat pipe at the joint(cable), and Rp refers to the average value of this resistance.Therefore, the average thermal conductance between con-ductor and interior of the heat pipe, per unit axial length,is

K = R^ = (Roi+lRHPr1 (44)

The thermal resistance over the oil film per unit axiallength is

In

(45)2nXol

wherero = outside radius of heat pipe5 = oil film thicknessXol = thermal conductivity of oil

The thermal resistance over the heat pipe consists of theradial resistance over the wall and the wick in the evapo-rator, the axial resistance in the vapour and the radialresistance over the wick and wall in the condenser.According to Chi [3], this heat-pipe resistance per unitaxial length is

2n I LeXc Leke

(46)

Substitution of the values of Table 4 gives the thermalresistance of the heat pipe: Rhp = 0.025 mK/W.

9.6 Thermal conductance between conductor and heatpipe in the experimental setup

In the experimental setup, a copper wire was coiled aroundthe heat pipe, while the thermocouple wires were wrappedbetween this copper helix (see Fig. 13). The thermal con-ductance kp between the conductor and the heat-pipe inte-rior, per unit axial length, is calculated according to thesimplified model as shown in Fig. 14:

R,Ro Rc

where

R _a + b + cVln {rjro)~\a |_ 2nXol \

'In {rjrb) ^ In fa/rji2nXol 2nXtc J

p n (rjre) | In (rc/r0)~\\_ 2nXol 2nXc J

a + b + c

a + b + c

The thermal resistance over the copper pipe is

„ In (rjrd)

and over the oil-impregnated paper

In (rf/re)

(47)

(48)

(49)

(50)

(51)

(52)

Substitution of the values presented in Fig. 14 in eqns. 46to 52 results in kp = 3.0 W/mK.

348 IEE PROCEEDINGS, Vol. 131, Pt. C, No. 7, NOVEMBER 1984