Numerical investigation of the thermohydraulic behaviour of a complete loop heat pipe

(1)

HAL Id: hal-01025872

https://hal.archives-ouvertes.fr/hal-01025872

Submitted on 11 Mar 2016

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Numerical investigation of the thermohydraulic

behaviour of a complete loop heat pipe

Benjamin Siedel, Valérie Sartre, Frédéric Lefevre

To cite this version:

(2)

Loop Heat Pipe

Benjamin Siedel,ValérieSartre

∗

,Frédéri Lefèvre

UniversitédeLyon,CNRS

INSA-Lyon,CETHILUMR5008,F-69621,Villeurbanne,Fran e

UniversitéLyon1,F-69622,Villeurbanne,Fran e

Abstra t

A ompletesteady-statemodelhasbeendevelopedtodeterminethethermohydrauli behaviourofaloopheat

pipe. Themodel ombinesanedis retizationofthe ondenserandthetransportlineswitha2-Ddes ription

oftheevaporator. Theseoriginalfeaturesenabletotakeintoa ountheatlossestotheambientandthrough

thetransportlines aswellasto evaluatetheparasiti heat uxthroughthewi kandtheevaporatorbody.

Thepresentnumeri alsimulationsmayimprovetheunderstandingofthephysi alme hanismsoperatingin

anLHPevaporator,andtheir ouplingwiththeotherpartsoftheLHP,andprovideguidan efortheLHP

design,aimingtoredu ethethermalresistan eofthesystem. The omparisonbetweenexperimentaldataof

aatdisk-shapedevaporatorfrom theliteratureandtheoreti alsimulationsvalidatestheproposed model.

Simulationsshowthesigni an eofheat ondu tionthroughtheliquidline. Additionally,resultsshowthe

majorinuen e oftheevaporation oe ientandof thewi k thermal ondu tivity ontheLHPoperating

temperature. Whenlongitudinalheat lossesarenotsigni ant,the ompetition betweentheparasiti heat

ux through thewi kand the heat transfer to the evaporation zone leadsto anextremum for whi h the

operatingtemperatureismaximal. With athermal ondu tiveevaporator asing,thelongitudinalparasiti

heatuxstronglyinuen estheLHPoperationaltemperatureandtheevaporatorenergybalan e.

Keywords: LoopHeatPipe, Modeling, Parametri study,Parasiti heatux, Evaporation oe ient

1. Introdu tion

Loop HeatPipes(LHP) are e ientheat transferdevi esableto passivelytransportlargeamountsof

heatandarethus onsideredasa ompetitivesolutionforele troni oolingappli ations. Thesetwo-phase

systems,developedin the1970's in Soviet Union,havealreadyproventheirreliableperforman ein many

spatialappli ationsandaretoday andidateforterrestrialappli ations.

∗

Correspondingauthor

(3)

pla e and of a ondenser, both elements being onne ted by separate vapour and liquid ow lines. The

passiveuid ir ulation,indu edbythe apillary pressurein theporousmedia,aswellastheuseoflatent

heat of vaporizationenablethe transport of high amountsof energy, even againstadverse gravity ee ts.

Extensivestudies havebeen madeto understand theoperatingprin iples of su h omplexsystemsand to

des ribethethermalandhydrodynami ouplingsoftheirdierentelements[1℄[2℄.

Inthepastde ade,alotofeortshavebeendevotedtothesteady-statemodellingofsu hsystemsfora

betterunderstandingoftheiroperationin ordertoimprovetheirdesign. Indeed,the hoi eoftheworking

uidandoftheloop'sgeometri alparametersplayanimportantrole intheLHP'soperationand anlead

tounexpe tedshutdownsortemperatureovershoots,puttingtheele troni devi e'sintegrityatrisk.

Most ofthese models an bedivided into two ategories. Arst groupof paperspresentsLHPglobal

operationmodelsdis retizingitintoseveralvolumeelementsorbasedonele tri alanalogiesanddes ribing

thewhole devi e asanodalnetwork. Thelinks betweenthenodesare representedby thermalresistan es

andtheenergybalan e equationisapplied toea hnode. Kayaetal. [3℄developedamathemati almodel

basedonthesteady-stateenergybalan eequationsatea h omponentoftheLHP.Theirsimulationsshow

satisfa tory a ordan ewith the experimental results. However, at low input powers, somedis repan ies

are pointedout; authors on lude that theheat ex hange with the ambient aswell asthe pressuredrops

in the ondenser need to be predi ted with more a ura y. Adoni et al. [4℄ developed a steady-state

thermohydrauli model tostudy theee tof ompensation hamberhard-llingaswellastheinuen eof

the bayonet on the operational hara teristi s of the LHP. The 1-D steady-state model of Chuang [5℄ is

basedontheenergybalan eequation,thermodynami relationshipsanddetailedheattransferandpressure

drop al ulations in the liquid, vapour and ondenser lines. This study des ribes extensively the LHP

operation in gravity-assisted onditions. Yet, heat transfer in the evaporator and the reservoir are not

pre isely des ribed. Launay et al. [6℄ presented an analyti al LHP global model, based on momentum

and energy balan e equations and thermodynami relationships. Two distin t losed-form solutions are

found for the various LHP operating modes, basedon a previous nodal numeri almodel. As well asfor

theother nodal models, themain drawba kis ana urate determination ofthe onsidered resistan es,in

parti ularthose des ribingthethermal linksbetweenthesaddle, thegroovesandthe reservoir. Baiet al.

[7℄ alsomodelled aLHPbasedonenergy onservation laws. Theirworkshowstheinuen e ofatwo-layer

ompound wi k and takesinto onsideration the liquid-vapour shear stressesin the ondenser. However,

ondu tioninthetransportlinewallsisnegle tedandsomeparametersdeningthethermalnetworkofthe

evaporatorhavetobeexperimentallyestimated. Su hmodelshavetheadvantageofdeterminingthemain

operationalparametersofthesystem,basedonlyontheinputheatux,thegeometri al hara teristi sand

theambient onditions. However,heat transferinsidethe evaporatorstru ture andthe ondenser are not

(4)

thevarious elementsof theevaporator anddes ribesthethermal andthe hydrauli statesof the wi k, as

well as the hara teristi s of the evaporation zone. Cao and Faghri [8℄ obtained analyti al solutions of

uid ow and heat transferin a wi kpartially heated with the onsideration of anevaporatinginterfa e.

Their work an provide a useful tool to deal with boiling in ipien e in the porous stru ture as well as

pressuredropinthewi k. Figusetal. [9℄developedaporenetworkmodeltodes ribetheporousstru ture

of the wi k. Their work is based on a pore size distribution and shows the fra tal displa ement of the

vapourfrontinside thewi kbeforethedeprime oftheLHP.Coquard[10℄further developedthismodelto

ombinema ros opi transportequationswiththeporenetworkapproa h. ZhaoandLiao[11℄ studiedthe

evaporativeheattransfer hara teristi sfroma apillarywi kheatedwithapermeableheatingsour eatthe

top. Theevaporating apillarymenis uswasmodelledataporelevelinordertodeterminetheheattransfer

oe ient. Severaloperating onditionswereobservedandadryingoftheporousstru tureo urredasthe

heat ux in reased, eventually leadingto the riti al heat ux. Ren et al. [12℄ developed anaxisymetri

two-dimensionalmathemati al model of awi k stru ture nearbya nand agroove. The apillary-driven

onve tionaswellastheinuen eoftheintera tionbetweentheoweldandtheevaporationinterfa eon

the urvatureofthemenis ihavebeentakenintoa ountinordertostudytheee tofseveralparameters

su haspermeability,porosityandporeradiusonthethermalperforman e. ChernyshevaandMaydanik[13℄

presented a omplete 3-D model of aatevaporator and studied the evaporationrate distribution in the

grooves,aswellasthestartofthewi kdry-out. Yet,allthesemodels annotpredi tthedevi eoperating

temperature for agiven heat ux and depend on experiments or other models to al ulate the reservoir

temperatureandthetemperatureoftheworkinguidenteringthereservoir.

The present work is an original way of ombining a omplete model with a ne des ription of the

evaporatorthermalandhydrauli states. LikeintheworkofRivièreetal. [14℄,thetransportlinesandthe

ondenser are dis retized and the onservation equations aresolvedfor ea h subvolumeof uid and ea h

subvolume of tube wall. This is one of the original features of the present model sin e heat ondu tion

throughthe tube walls wasalways negle tedin the previousworks, although itmight havean impa ton

theLHPoperation,inparti ularforlowinputpowers. Additionally,a2-Ddes riptionoftheheatandmass

transferin the evaporator is ondu ted to a urately predi t the heat ux distribution in the evaporator

asing,intheporouswi kandattheliquid-vapourinterfa e.

2. LHP GlobalModel Formulation

2.1. LHPdes ription andassumptions

As shown in Figure 1, a LHP onsists of an evaporator, a vapourline, a ondenser, a liquid line and

(5)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condenser Liquid line Vapour line Evaporator Reservoir Wi k Vapourgrooves

Figure1:S hemati ofaLHPwithaatdisk-shapedevaporator

Thesteady-statemodelpresentedhereisbasedonmomentum,massandenergy onservationequations

andonthermodynami relationships. Themajorassumptionsare:

(a) Theloopoperatesat steady-state.

(b) The wi kis fully saturatedof liquidand evaporationtakespla e at the liquid-vapourinterfa ein the

vapourgrooves.

( ) The uid is onsidered in ompressible be ause itsvelo ityis mu h lowerthan thesound speedin the

samemediuminatypi alLHPoperation.

(d) Thetemperaturedieren ebetweentheinner andtheoutersurfa es ofthetubesisnegle ted.

(e) The heatsink temperature andtheexternal heattransfer oe ientare onsidereduniform alongthe

ondenser.

2.2. Vapour line,liquid lineand ondenser

The ondenserandthetransportlinesaredis retizedintosmallelementsofvolumeasshowninFigure2.

Todeterminethe wall temperature

T

wall

r

Figure2: Transportlinesdis retization

where

T

out

istheambientorheat sinktemperature,

h

out

and

h

in

are theheat transfer oe ientsoutside

andinsidethetuberespe tively,

p

istheperimeterand

A

the ross-se tionalareaofthetube. Insingle-phaseow,theuidtemperature

T

f isgivenby:

dT

f

dz

=

h

in

p

in

(T

wall

−

_T

f

)

˙

m

f

c

p,f (2)

Intwo-phaseow,theuidisat saturationtemperature

T

sat

andthevariationoftheliquidmassowrate

isexpressedby:

d ˙

m

l

dz

=

h

in

p

in

3 +

dP

dz

v

x

3

(7)

Assumingthevapourandliquidvelo itiesareequal,thevoidfra tionisgivenby:

ε =

1 +

1 − x

x

ρ

v

ρ

l

−1

(8)

As suggestedby Thome[16℄, the two-phaseowis hara terized onsidering a homogeneouspseudo-uid

(7)

ρ

h

= ρ

l

(1 − ε) + ρ

v

ε

(9)

Themomentum pressuregradientperunit lengthisthen:

dP

dz

mom

=

m

˙

f

A

d

dz

˙

m

f

ρ

h

A

(10)

whereasthehydrostati termisdenedby:

dP

dz

stati

= ρ

h

g

dH

dz

(11)

where

H

isthealtitudeofthe onsideredelement.

Theheattransfer oe ientwiththeambientisgivenbyChur hillandChu [17℄forfree onve tionon

thesurfa eofanisothermal ylinder:

h

amb

=

λ

air

D

out







0.60 +

0.387Ra

1

6

D

1 + (0.559/P r)

16

9

27

8 





2

(12)

This orrelationisvalidforRayleighnumbers

Ra

D

lowerthan

10

=

D

z

Re

D

P r

(13)

Theheat transferbetweentheuidandthetubewallisthengivenby[18℄:

Gz

D

6

20 h

in

= 4.36

λ

l

D

in (14)

Gz

D

> 20

h

in

= 1.86

Re

D

P r

L/D

1

3 µ

f,T=Tf

µ

f,T=T wall

0.14 λ

l

D

in (15)

Foraturbulentow,theentran elengthismu hsmallerandfullydeveloped onditionsareassumed:

h

in

= 0.023

λ

f

D

in

Re

4

5

D

P r

1

3

(16)

In the ase of two-phase ow, a ne modelling of the heat transfer asso iated with ondensation an be

foundintheworkofMis evi etal. [19℄. Thismodeltakesintoa ountthetransientbehaviouroftheliquid

shapeinside thetubedueto the apillary ee ts. However,su h anapproa hisdi ultto introdu ein a

ompletesteady-statemodelofaLHP.Thus, anotherapproa hisfollowed: theSoliman'smodiedFroude

numberdeterminestheowregimeasafun tion ofthedimensionlessliquidReynoldsandGalileonumbers

(8)

with

Ga =

gD

3 _ρ

2

l

µ

2

l and

Re

l

=

4 ˙

m

tot

(1 − x)

πDµ

l (19) andwhere

χ

tt

istheMartinelliparameterdenedas[21℄:

χ

tt

=

s

(dP/dz)

l

(dP/dz)

v

≈

1 − x

x

0.9 _ρ

v

ρ

l

0.5 _µ

l

µ

v

0.1

(20)

ForFroude numberslowerthan 10,theowis assumedstratied andthe orrelationofJasterand Kosky

[22℄isusedtodeterminetheheattransfer oe ient:

h

in

= 0.728ε

3

4 gh

lv

λ

3

l

(ρ

l

−

_ρ

v

) ρ

l

D

in

µ

l

(T

f

−

_T

wall

)

0.25

(21)

Theowisassumedannularotherwiseand the orrelationgivenbyAkersetal. [23℄isused:

if

Re < 5 × 10

5 _h

in

= 5.03

λ

f

D

in

Re

1

3

eq

P r

1

3

l (22) if

Re > 5 × 10

5 _h

in

= 0.0265

λ

f

D

in

Re

4

5

eq

P r

1

3

l (23)

wheretheequivalentReynoldsnumberfortwo-phaseow

Re

eq

isdeterminedfromanequivalentmassow

rate:

˙

m

eq

= ˙

m

tot

"

(1 − x) + x

ρ

l

ρ

v

1/2

#

(24)

2.3. Determinationofthe mass owrate

In a rst approa h, the heat losses to the ambient, the heat transferred by ondu tion through the

transportlines aswell asthesidewallparasiti heatlosses arenottakeninto a ount. Asimplied global

energybalan eoftheevaporatorgivesthenthefollowingequation:

Q

in

= Q

ev

+ Q

sen

+ Q

sub

(25)

where

Q

in

istheheatdissipatedbythedevi ethathastobe ooledbytheLHP.Thetermsontheright-hand

sideoftheequation orrespondtothelatentheatofvaporizationoftheuidatthewi ksurfa e,thesensible

heatprovidedtotheliquidin thewi k,andthesub oolingdueto thereturnofliquidfromthe ondenser.

ConsideringtherelationshipsbetweentheseheattransferratesandthemassowrateinsidetheLHP:

Q

ev

= ˙

m

f

h

lv (26)

Q

sen

= ˙

m

f

c

p,l

(T

gr

−

_T

res

)

(27)

Q

sub

= ˙

m

f

c

p,l

(T

res

−

.

T

res,in

is alsodetermined usingthe transportlines model

whilethevalueof

T

gr

issolvedusinga2-Dmodeldes ribedin thenextse tion. Themassowrateisthen

al ulatedby ombiningEq. (25)to (28):

˙

m

f

=

Q

in

h

lv

+ c

p,l

(T

gr

−

_T

res,in

)

(29)

2.4. Heatandmasstransferin thewi k

Intheliterature,LHPmodelsusually onsiderheattransferthroughthewi kasauniform1-D

ondu tive- onve tivetransfer,assumingan equivalentthermal ondu tivity ofthewi k. Inorderto determine

a u-ratelyheatandmasstransferintheporousmedia,a2-Dnitedieren edes riptionofthewi kisundertaken

inthis model. As shownin Figure 3,asmall elementof theevaporatorhasbeen hosenfor thisstudy,

lo- atedbetweenthe entre ofthevapourgrooveandthe entre ofthenearbyn,betweentheliquid-vapour

interfa einthereservoirandtheuppersurfa eoftheevaporatorwall. Thefollowingassumptionsaremade:

(a) Heatandmasstransferaretwo-dimensional.

(b) Evaporationonlyo ursatthesurfa eoftheporousstru turenearbythevapourgroove.

( ) Lo alliquidandwi ktemperaturesare onsideredequal.

(d) Gravitationalee ts arenegle ted.

(e) Heat lossestotheambient,throughthetransportlinesand parasiti heatux throughtheevaporator

asingarenegle ted.

(f) The reservoir ontainsatwo-phase uid. As it is lo ated abovethe evaporator, aheight of liquid

H

l

sitsonthetopofthewi kstru ture.

BasedonthetotaluidinventoryoftheLHP,assumingthewi kisfullysaturatedofliquidand al ulating

thevoidfra tionalongthetransportlinesandthe ondenser,itispossibletodeterminetheheightofliquid

H

l

in thereservoir.

Todes ribetheliquidowinsidetheporousstru ture,Dar y'slawis onsidered:

u = −

K

p

ε

3 37.5 (1 − ε)

2

(31) where

r

p

is thepore radiusand

ε

istheporosityof thewi k. Consideringthe ontinuityequation andEq. (30),thepressureeldinthewi k anbewritten:

(10)

y

Figure3: Heatandmasstransferinthewi k

Thefollowingboundary onditionsare onsideredattheboundariesAtoF:

AandC:

∂P

∂x

= 0

(33) B:

P = P

res (34) D:

∂P

∂y

= 0

(35) F:

P = P

gr (36)

As frontiers A and Care symmetryaxes, theliquid velo ities haveno

x

- omponent. Atthe boundary D, theimpermeability onditionis applied. Thepressurein the grooveisset by thethermodynami stateof

thelatter

P

gr

= P

sat

(T

gr

)

.

Inorderto al ulatethetemperatureeld, ondu tionintheporousstru tureand onve tiveheatingof

theliquidowingthroughthewi kare onsidered,leadingto thefollowingenergyequation:

λ

e

∇

2 _{T = ρ}

l

c

p,l

∇

_{(uT )}

(37)

Several models an be found in the literature to estimate the ee tive thermal ondu tivity

λ

e

of a

sintered porous stru ture lled with liquid. Based on the work of Singh et al. [25℄ who ompared their

(11)

λ

e

= λ

l

λ

l

λ

w

−(1−ε)

0 .

59

(38)

Theboundary onditionsareasfollows:

AandC:

∂T

∂x

= 0

(39) B:

−

λ

e

∂T

∂y

=

λ

l

H

l

(T − T

res

)

(40) E:

−

λ

ev

∂T

∂y

=

Q

in

A

w (41) F:

h

ev

(T − T

gr

) = −h

lv

u

y

ρ

l (42) G:

−

λ

ev

∂T

∂y

The sides of the onsidered region (dashed lines) are assumed adiabati be ause of the symmetry. The

thi knessof theliquidlayerinthe reservoirissupposed to besmall enoughtonegle t the onve tiveheat

transfer. Thus aFourierboundary onditionis appliedwith aheat transfer oe ientdened bythe1-D

heat ondu tionin theliquid. The onve tiveheattransfer oe ientinthegroove

h

gr

issettoa onstant

value onsideringnatural onve tion. Attheouterwallsurfa e,axedheat ux

Q

in

/A

w

isapplied, where

A

w

isthe ross-se tionalareaofthewi k.

Athermal onta tresistan ebetweentheporousstru tureandtheevaporatorwallhastobe onsidered.

Choietal.[26℄studiedthethermal hara teristi sofseveralvapor hannelgeometries. Basedontheirwork,

a onstantgoodthermal onta tisassumedandthe onta tresistan eisequalto

5.10 −5

_K·m

2 _·

_W

−1

.

sat

2πRT

sat

M

−0.5

1 −

P

sat

2ρ

v

h

lv

(45) where

a

ev

istheevaporation oe ient. Inthe aseoftheevaporationofathinliquidlm,theevaporation

oe ientisdenedas theratioofthea tualevaporationratetoatheoreti almaximalphase hangerate.

A oe ient equal to unity des ribesa perfe t evaporation while a lowervalue represents an in omplete

evaporation. Inthe aseofwater,valuesvaryingfrom0.01to1aresuggestedintheliterature[28℄.

2.5. Solving pro edure

The ompletesolvingpro edureispresentedinFigure4. Foragivenheatux,thethermalandhydrauli

(12)

Cal ulationoftemperatures

T

f

,pressures

P

f

,massvapour

qualities

x

andheat trans-fer oe ients

h

in and

h

out

inthetransportlinesand

the ondenser(Eq. 1to24 ) Determinationofthemass

owrate

m

˙

f

(Eq. 29)

Cal ulationofpressure

andvelo ityeldsinthe

wi k(Eq. 30and32 )

Cal ulationofthetemperature

eldinthewi k(Eq. 37)

Estimationofthegroove

temper-ature

T

Plotoftheoperating urve yes no yes no no yes

(13)

perature

T

gr

,themaximumwalltemperatureintheevaporator

T

ev

,thereservoirtemperature

T

res

andthe

massowrate

m

˙

f

. Thepressureattheentran eofthevapourlineisthesaturationpressure orresponding

tothevapourtemperaturein thegroove

T

gr

. ThetubewalltemperaturesaredeterminedsolvingEquation

(1)with theThomasalgorithm(also knownasTridiagonalMatrixAlgorithm), using

T

ev

and

T

res

at ea h

endofthetransportlinesasboundary onditions. Thentheuidpressures

P

f

,themassvapourqualities

x

, theinnerheat transfer oe ients

h

in

aswellastheuidtemperatures

T

f

are omputedstepbystepwith

aniterativemethod, using the approa h presented in 2.2, until ea h al ulated parameterhas onverged.

Theenergybalan eattheevaporatorenablesto re al ulatethemassowrate

m

˙

f

usingEquation(29).

Themodeldes ribingheatandmasstransferintheevaporatoristhen omputed. Theinputparameters

are the mass ow rate

m

˙

f

, the reservoir pressure

P

res

and its orresponding saturation temperature

T

res

as well asthe temperature of the liquid entering the reservoir

T

res,in

. First, the pressure eld and then

the temperature eld in the porous stru ture are al ulated, using the Thomas algorithm alternatively

in both dire tions. The groove temperature

T

gr

is estimated with a shooting method.

T

gr

is iteratively

modieduntiltheheat uxin theevaporationzoneisequalto

m

˙

f

h

lv

. Whentheerrorson

T

gr

and

T

ev

are

negligible,the ompletethermo-hydrauli stateoftheLHPis onsidereddeterminedandanotherinputheat

uxis omputed,untilthe omplete hara teristi urveis al ulated. The omputationaltimerequiredto

al ulatea14-points hara teristi urveisofabout

8 min

.

2.6. Modellingof the heattransferthrough the asing

The model presented in Se tion 2.4 does nottakeinto a ountthe heat losses from theevaporator to

theambientnorthethermal ondu tionthroughtheevaporator asingtothereservoirandtothetransport

Q

in

−

_Q

tube,v

−

_Q

tube,l

−

_Q

amb

h

lv

+ c

p,l

(T

gr

−

_T

res,in

)

(47)

−

_λ

ev

∂T

∂y

= (Q

in

−

_Q

tube,v

−

_Q

wall

)

in Eq. (48).

Q

amb

is equaltotheheat lossestotheambient.

Q

wall

is

thepartoftheheat transferred throughtheevaporatorsidewallthatis notgiven ba kto thewi kdue to

onve tiveheattransferbetweentheliquidintheporousmediaandthesidewall.

Q

wall

(14)

longitudinalheatlossesin thefollowing. Apartof

Q

wall

isthendissipatedto theambientwhiletherestis

givento thereservoir. Totakeintoa ountheat losses,the 2-Dmodel presentedin Figure 3is omputed

usingEq. (46)to (48).

The estimationof both

Q

amb

and

Q

wall

requires anadditionalmodel. Obviously, a3-D des riptionof

theentireevaporatorwouldgivethemosta urateresults. However,in ordertoredu e the omputational

time, a 2-D approa h has been hosen in this study (Fig. 5). The number of grooves to be modelled is

determinedsothatedgeee tsarenegligibleatthe entreofthegroovelo atedattheoppositesideofthe

sidewall. The dashedline isthus assumedto beanadiabati boundary. Thevapour zonein thereservoir

is onsidered at ahomogeneoustemperature

T

res

. Contrarilyto themodelpresentedin Figure 3,the2-D

heat ondu tionequation issolvedintheliquidto takeintoa ounttheheattransferwith the asing. We

assumethattheliquidthi kness

H

l

inthereservoirissmallenoughtonegle t onve tionphenomenainside

theliquid. Theliquid-vapourinterfa eI isset to

T

res

. A perfe t onta tisassumed betweenthe body of

theevaporatorandthewi k.

Atthe onta tsurfa ebetweentheevaporator asingandthevapour,twodistin tsituationsmayo ur:

if the vapour is older than the wall, the liquid evaporates in the reservoir whereas ondensation of the

vapouro urs otherwise. In the aseof evaporationin thereservoir,heat transferis greatlyenhan ed at

thetriple line, at the onta tbetween thewalland the liquid-vapourinterfa e. Theheat transfer anbe

determined using anevaporationheat transfer oe ient al ulated with Eq. (45). In this situation,the

onve tiveheattransferbetweenthewallandthevapourisnegle tedandanadiabati boundaryisassumed.

Theheatdissipatedfromthereservoirtopsurfa etotheambient

Q

amb,top

aswellastheheatlossesthrough

theliquidline arealsoduetotheheat ondu tionin the asing. Thus,theboundary onditionJis:

−

_λ

ev

∂T

∂y

=

4 (Q

amb,top

+ Q

tube,l

)

π (D

2

wall

−

_D

2

w

)

(49)

Inthe eventof ondensation ofthe vapour onthe reservoirwall, the heat transferalongthe wallis xed

to

10

4 _W·m

−2

· _K

−1

, based onNusselt's lmwise ondensationtheory [18℄. With su ha high heat transfer

oe ient,theheatdissipatedfromthereservoirtopsurfa eandthroughtheliquidlineis omingfromthe

reservoirandnotfromthe asing,asintheprevious ase. Therefore,theboundaryJis onsideredadiabati

inthat ase.

The boundary onditions (33) to (36) and (42)to (44) are similar to those of themodel presented in

Figure3. Ontheevaporatorbottomsurfa e,a onstantheat uxisset:

−

_λ

ev

∂T

∂y

=

Q

in

−

_Q

tube,v

A

w (50)

Asshownin Figure5,apartof theheat ondu ted throughtheevaporator body

Q

wall

isdissipatedto

theambientwhiletherestis transportedto thereservoirand throughtheboundaryJ.

Q

amb

is al ulated

(15)

h

amb I J

(Q

in

−

_Q

tube,v

) /A

w wi k liquid vapour

Q

w all

x

y

meanpore radius. Theporous wi kis embedded in apure opperevaporator (

λ = 398W.m

−1

_.K

−1

),

10 mm

thi k and with an a tive zone diameter of

30 mm

. The vapour and liquid lines, of

2 mm

internal diameter,are

150 mm

and

290 mm

longrespe tivelyand alsomadeof pure opper. Then-and-tube ondenser,

50 mm

long,dissipatesheatbyfor ed onve tionofairatambienttemperature (i.e.

22 ◦

_C

). A straight-tube equivalent ondenser is simulated with an external heat transfer oe ient

arbitrarily hosenequalto

2.6 kW·m

−2

· _K

−1

,a ordingto theexperimental results, onsideringan outside

diameterof

2.4 mm

forthetubes. Fifteengrooveswithasquare ross-se tionalareaof

1 mm

2

aremodelled

and thewallthi knessof theevaporator is taken equalto

2 mm

. A reservoirheight of

4 mm

is hosento providethevolume ne essaryforthetotaluidmassof

5 g

in theentire heatinputrange. An evaporation oe ientequalto

0.02

is hosentottheexperimentaldata. Sin etheloopwasthermallyinsulatedwith breglass,heatlossestotheambientarenegle ted. Furthermore,asanO-ringsealpreventsheatfrombeing

(16)

shownin Figure6. A good agreementisobtainedfortheentire inputpowerrange,althoughforhigh heat

loads themodel tendsto predi t higherevaporatortemperaturesthanthe experimental data. The

exper-imental datas attering showssomenon repeatabilityof theLHPoperation. The meanerrorbetweenthe

simulationandexperimentaldataisof

2.6 K

,whi hisonthesameorderofmagnitudeastheexperimental resultss attering.

10

20

30

40

50

60

70

55

60

65

70

75

80

85

90

95

100

105 Q

in

(W)

T

ev

(

°

C)

T

model

T

exp

VCM FCM

Figure6:ComparisonbetweenthemodelanddatafromSinghetal. [29℄

One an learlyseethetwodistin toperatingmodes,knownasvariable ondu tan emode(VCM), up

to about

40 W

, where the two-phase length anvary in the ondenser, and the xed ondu tan e mode (FCM) forwhi h theend of the ondensationtakespla e at the veryend of the ondenser orevenin the

liquid line. In the latter, the evaporator temperature varies almost linearlywith the heat load and at a

higherratethanin VCM.

The evaporation oe ient an have a signi ant inuen e on the heat transfer in the wi k lose to

thevapourgroovesand thens. Indeed,alowevaporation oe ientmoderatestheevaporationrateand

attenstheevaporationrateproleatthewi ksurfa ein onta twiththegroove. Figure7presentstheee t

oftheevaporation oe ientontheevaporator temperature

T

ev

. De reasing

a

ev

stronglyae tstheLHP

(17)

keyparameteroftheloopheatpipemodelling,butpresentlyitsvalueisnotwellreferen edintheliterature.

Inthe followingparametri study, simulationswill be ondu ted with a onstant oe ientequalto

0.02

based on the validation results. Additionally, to ensure a good evaporator temperature homogeneity, a

groovedwallmadeof opperwillbesimulated(

λ = 398 W·m

−1

· _K

−1

).

10

20

30

40

50

60

70

50

60

70

80

90

100

110 Q

in

(W)

T

ev

(

°

C)

a

ev

=0.01

a

ev

=0.02

a

ev

=0.1

a

ev

=1

Figure7:Inuen eoftheevaporation oe ientontheLHPoperatingtemperature

4. Resultsand dis ussion

4.1. Thermal ondu tion inthe vapour andliquid lines

Figure8presentsthe omparisonofdierentmaterialsforthevapourandliquidlinesandthe ondenser,

negle tingthelongitudinalheatlosses

Q

wall

aswellasheatlossesfromtheevaporatortotheambient

Q

amb

.

Invariable ondu tan emode,high ondu tivematerialsleadtobetterLHPperforman e. Indeed,alarger

partof the heat to be dissipated is ondu ted through the lines and this leadsto a ooling of the LHP,

sin eahighertemperatureofthetransportlinewallindu esmoreheatlossestotheambient. Athighinput

powersthedieren eisnotsigni antandtheheat ondu tionthroughthetransportlinesdoesnotstrongly

inuen etheLHPoperation.

At low input powers, the mass ow rate in the tubes is extremely low. In addition, a large part of

(18)

10

20

30

40

50

60

70

55

60

65

70

75

80

85

90

95

100

105 Q

in

(W)

T

ev

(

°

C)

glass

steel

aluminium

copper

Figure8: Inuen eofthetransportlinesthermal ondu tivityontheLHPoperatingtemperature

exit of the ondenser (

x = 0.2 m

) is lose to the heat sink temperature (Fig. 9). The vapour superheat islimited at the exit ofthe vapour groovesandthe vapour line wallis ooleddown by theheat losses to

theambient. Therefore,thevapouralready ondensatesinthevapourline (

x = 0.02 m

)before entering in the ondenser. Then, the uid temperature is equalto the saturation temperature and the high internal

heattransfer oe ienttendstoimposethetemperatureofthevapourtothewalluntiltheentran eofthe

ondenser. In thepartof the ondenser lledwithliquid, boththewalland liquidtemperaturesde rease

torea htheheatsink temperature.

In the aseof a opper liquid line, a signi ant length of the tube wallis heated by ondu tion from

the reservoir asing. The liquid returning to the ompensation hamber is then at ahigher temperature

T

res,in

, providing averylow sub ooling. Sin e the liquid returningto the ondenser is sub ooled, a large

amountof heat is ondu ted from the evaporatorthrough theliquid line to be dissipated in onta twith

theliquidowinginthetube. Thisee tisin reasedinvariable ondu tan emodebe ausetheliquidexits

the ondenserat atemperature lose totheheat sink temperature, thus providing alargersub ooling. In

the aseof aglasstube,heat ondu tion in thetransport line wallis almost negligibleandtheliquid line

wall is older thanwith a opperline. As aresult, theheat losses to the ambientde rease and theLHP

operatingtemperatureishigher.

Athighheatloads,thevapourfrontisattheveryendofthe ondenser(Fig.10). Thus,theliquidlength

(19)

0

0.1

0.2

0.3

0.4

10

20

30

40

50

60

70

80 Length (m)

T (

°

C)

T

wall,glass

T

f,glass

T

wall,copper

T

f,copper

Vap. Line Cond. Liq. Line

Figure9: Temperaturealongthetransportlinesfor

Q

in

= 10 W

de reaseof thewalltemperatureat theend ofthe ondenseris dueto thesharpvariationofheat transfer

oe ientbetweentheuidandthetube. Atthebeginningofthe ondenser,the ondensationphenomena

tendsto imposethetemperatureofthevapourtothewall. Attheend ofthe ondenser,theheat transfer

oe ientwith theliquid ismu h smaller,and thus theheat sink tendsto imposeits temperature tothe

wall. Asaresult,thetemperatureoftheliquidde reasesattheendofthe ondenser. Intheliquidline,the

wall and theliquid owing in the tube rea h approximatelythe sametemperatureand, asa onsequen e

of heat losses to the ambient, the temperature de reases along the liquid line. The liquid is heating up

beforeenteringthereservoir,duetoheat ondu tionfromthereservoir asing. Thus,thewallandtheliquid

temperaturesare losertothereservoirtemperaturethanatlowheatloadsandthetransportlinematerial

hasaminorinuen e, theliquidsub oolingbeingalmost onstantwhateverthe ondu tioninthetubes.

4.2. Wi kthermal ondu tivityandthi kness

Heat uxesthroughthewi kbeingsigni antfortheLHPoperation,severalwi kmaterialshavebeen

ompared, negle ting the heat transfer through the evaporator body and the heat losses to the ambient

(Fig.11).

Theresultsshowthatinthevariable ondu tan emode,ade reaseofthewi kthermal ondu tivityleads

(20)

0

0.1

0.2

0.3

0.4

70

75

80

85

90

95

100

105

110 Length (m)

T (

°

C)

T

wall,copper

T

f,copper

Vap. Line Cond. Liq. Line

Figure10: Temperaturealongthetransportlinesfor

Q

in

= 70 W

materialssu hasplasti sor erami s,anoppositeee tisobserved. Athighheatloads,theee tofthewi k

materialislesspronoun ed. Su h anextremumisthe onsequen eoftwooppositephenomena. De reasing

thewi k ondu tivityredu esthepartofheatpassingthroughtheporousstru ture,thusleadingtoalower

transversalparasiti heatux. Thesub oolingof theliquidenteringthereservoiris thenredu edandthe

LHPoperatesatalowertemperature. Atthesametime,amore ondu tiveporousstru tureenhan esthe

evaporationat thewi ksurfa ein onta twith thevapourgroove. Whentheevaporationheattransferat

the onta t line between the n, the groove and the wi k is limited (small valueof

a

ev

) or when a high

thermal onta tresistan eexistsbetweenthenandtheporousmaterial,heathastobe ondu tedthrough

alonger path in the wi k in order to beevaporated. In that ase,a lower ondu tivity ande rease the

evaporationrateandleadsto higherLHPtemperatures.

Figure12presentsthenon-dimensionalheattransferrates(denedastheratiobetweentherateofheat

owanditsmaximalvalue)intheevaporatorasafun tionoftheporousmaterialthermal ondu tivity. An

extremumisfoundfor athermal ondu tivity

λ

w

of

10 W·m

−1

· _K

−1

, i.e. anee tivethermal ondu tivity

ofthewi k

λ

e

equalto

2 W·m

−1

· _K

−1

. Abovethat value,theevaporationrate

Q

∗

ev

is enhan edbyamore

ondu tivewi kbe ause ondu tion in theporous stru ture is improved near theevaporation zone. This

ee t over omes the apa ity of the wi k to transfer the total parasiti heat ux

Q

∗

par

(21)

10

20

30

40

50

60

70

50

60

70

80

90

100

110 Q

in

(W)

T

ev

(

°

C)

λ

=200 W/m.K

λ

=90 W/m.K

λ

=10 W/m.K

λ

=2.5 W/m.K

λ

=0.4 W/m.K

Figure11: Inuen eofthewi kmaterial ondu tivityontheLHPoperatingtemperature-

Q

wall

and

Q

amb

negle ted

by ondu tion. Forlowerthermal ondu tivities, heat leaksthroughthe porous material are onsiderably

redu ed. Thus, heatispreferentiallydissipatedbyevaporationat thewi ksurfa eandthis leadsto better

performan e.

The ompetition betweenthese two ee ts isparti ularly observedwhen the evaporationrate lose to

the nis limited. This is onrmed by Figure 13 where the non-dimensionalevaporation rateis plotted

as a fun tion of the wi k material ondu tivity, for several evaporation oe ients

a

ev

. With a higher

evaporation oe ient,alargerpartofthetotalevaporatedmassowratetakespla eatthesurfa eofthe

liquidlm losetothen. Insu ha ase,thewi k ondu tivityhaslessinuen e ontheevaporationrate

andtheextremumis foundformaterials havingahigh thermal ondu tivity. Therefore,thewi k thermal

ondu tivityshouldbeaslowaspossibletoblo ktheparasiti heat uxthroughthewi k.

The same simulations have been realized onsidering longitudinal heat losses and heat losses to the

ambient. Non-dimensional heat transfer rates are presented in Figure 14. When the heat losses to the

ambientandthroughtheevaporator asing annotbenegle ted,noextremum anbefound.

When the thermal ondu tivity of the porous stru ture is low, the evaporation transfer rate and the

sensibleheatgiventotheliquidinthewi karelower. Atthesametime,althoughtheheattransferthrough

the wi k de reases, parasiti heat losses in rease due to the ondu tion in the evaporator asing. These

phenomenaleadtoamu hhigheroperationaltemperaturewhenthewi kthermal ondu tivityislow.

(22)

0.2

0.5

1

2

5 10 20

50

200

0.7

0.75

0.8

0.85

0.9

0.95

1 λ

_w

(W/m.K)

Q*

ev

Q*

tube,l

Q*

par

Figure12:Inuen eofthewi kthermal ondu tivityontheLHPperforman e-

Q

wall and

Q

amb negle ted

0.5 1

2

5 10 20

50

200

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1 λ

(W/m.K)

Q*

ev

a

ev

=0.02

a

ev

=0.1

a

ev

=1

Figure13:Inuen eoftheevaporation oe ienton

Q

∗

(23)

0.2

0.5

1

2

5 10 20

50

200

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 λ

_w

(W/m.K)

Q*

ev

Q*

amb

Q*

wall

Q*

tube,v

Q*

tube,l

Q*

sen

Q*

par

Figure14: Inuen eofthewi kthermal ondu tivityontheLHPperforman e onsideringheatlosses

longitudinalparasiti heatlosses anbenegle ted(insulatedLHPwithlow- ondu tivityevaporator asing),

onehasto avoid sele tingamaterialin amedium ondu tivityrange forthewi k. Theuseof abiporous

wi k ora se ondarywi k with lower ondu tivity analso help redu ing heat leaks withoutloweringthe

evaporationrate. Likewise,thi kerwi ks anpreventheatfrompassingthroughtheporousstru turetothe

reservoir(Fig. 15)andare lesssus eptibleto deprime in the aseof anevaporationfrontdispla ementin

thewi k. However,a ompromisehas to befound, sin eagreater wi kthi knessleadsto higherpressure

dropsand anbelimitedbythespa eavailablefortheintegrationoftheevaporator. Whenthelongitudinal

heatlossesaredeterminant,oneshouldrather hooseawi kmadefromagoodthermal ondu tivematerial

toensureagood evaporationrate.

4.3. Heattransferinthe evaporator

Figure16presentsthedierent omponentsoftheheatux intheevaporatordividedbythetotalheat

load

Q

in

,without onsideringlongitudinalheatlosses. The

2 mm

thi kevaporator asingismadeof opper andtheloopisnotinsulated. Ani kelwi kis onsidered,withanee tivethermal ondu tivity

λ

e

equalto

5.4 W·m

−1

· _K

−1

. Morethan

90 %

oftheheatisdissipatedbyevaporation

Q

ev

while

2

to

10 %

istransferred throughtheliquidline

Q

tube,l

. Theparasiti heat ux throughthewi k

Q

par

(24)

10

20

30

40

50

60

70

55

60

65

70

75

80

85

90

95

100

105 Q

in

(W)

T

ev

(

°

C)

e

w

=1 mm

e

w

=3 mm

e

w

=5 mm

Figure15: Inuen eofthewi kthi kness-

Q

wall

and

Q

amb

negle ted

evaporationrateisenhan edand heatlossesthroughtheliquidlinearelesssigni ant.

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

80

90

100 Q

in

(W)

Q/Q

in

(%)

Q

ev

Q

tube,v

Q

tube,l

Q

sen

Q

20 %

).

Q

amb

represents

theheatlossestotheambientfrom thereservoirandtheevaporatoranditsvalueisupto

20 %

ofthetotal heatux. Thetotalparasiti heatux

Q

par

ismoreimportant onsideringtheheattransferby ondu tion

throughtheevaporatorbody(

Q

wall

= 2 − 5 %

)andleadstoavalueof

2

to

15 %

. Heatlossestotheambient are less dominant for high heat loads whereas the total parasiti heat ux and the evaporation rate are

enhan ed.

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

80 Q

in

(W)

Q/Q

in

(%)

Q

ev

Q

amb

Q

wall

Q

tube,v

Q

tube,l

Q

sen

Q

par

Figure17: HeattransferintheLHP onsideringheatlosses

4.4. Evaporator body materialandthi kness

Two asing materialswith abody thi knessequalto

2 mm

havebeentested in orderto determinethe inuen e ofthe outer wall thermal ondu tivity onthe evaporatortemperature(Fig. 18). Additionally, a

opper evaporator with a thinner wall (

e

wall

= 1 mm

) has been simulated and all the results have been

ompared to the modelling without heat losses. Indeed, the parasiti heat ux from the saddle to the

reservoir,aswell as the heat ex hange with theambient anbe strongly dependent onthese parameters.

Theother geometri alandambient hara teristi softhemodelling arepresentedin Se tion3.

The hoi eof averylowthermal ondu tivity materialsu h asPEEK(

λ

wall

= 0.25 W·m

−1

· _K

−1

)

(26)

10

20

30

40

50

60

70

0

20

40

60

80

100

120

140

160

180 Q

in

(W)

T

ev

(

°

C)

no longitudinal parasitic heat flux

copper wall (

λ

= 398 W/m.K)

thinner copper wall

PEEK wall (

λ

= 0.25 W/m.K)

Figure18: Inuen eoftheevaporatorsidewall ondu tivityandthi kness

resultsarevery losetothesolutionwithout onsideringthelongitudinalparasiti heatux. However,heat

lossesto theambientde rease theevaporatortemperature. The useofavery ondu tivematerial su h as

opper(

λ

wall

= 398 W·m

−1

· _K

−1

) stronglyae ts the LHPoperationaltemperature, be ausetheparasiti

heatuxthroughtheevaporatorsidewallisin reased. Thisphenomenaismorepronoun edwhenthe asing

isthi ker,sin ethe ross-se tionalareaforthe ondu tiveheattransferislarger.

4.5. Heatlossestothe ambient

AlthoughLHParenotinsulatedin usualappli ations, one anndin theliteraturemanyexperiments

withthetubesorthewhole loopinsulated. Thepurposeofthis istominimizeheat lossesto theambient,

often di ult to evaluate, in order to ease the model validation or to hara terize the LHP performan e

withouttakingintoa ounttheenvironmentinuen e. Figure19presentsthe omparisonoftheevaporator

temperaturewithandwithoutglass-breinsulationontheevaporator-reservoirstru ture. Theresults learly

show that without insulation, the LHP performan eare enhan ed. Indeed, whateverthe heat load, heat

lossestotheambient oolthesystemand anbeofgreatimportan eintheLHPheatbalan e. Inthe aseof

alow ondu tiveevaporator asing,theinuen eoftheinsulationislessnoti eablebe ausetheevaporator

bodyitselfpreventsheatfrombeing ondu tedtotheevaporatorsideandthusde reasesheat lossestothe

(27)

10

20

30

40

50

60

70

60

80

100

120

140

160

180 Q

in

(W)

T

ev

(

°

C)

insulated copper

not insulated copper

insulated PEEK

not insulated PEEK

Figure19: Inuen eoftheinsulationontheLHPoperatingtemperature

5. Con lusion

In this study, a omplete model of LHP has been presented. It ombines a 2-D des ription of the

evaporatorhydrauli andthermalstateswithanedis retization ofthetransportlinesandthe ondenser.

These original features enable to takeinto a ountheat losses to theambient and throughthe transport

linesaswell asto evaluate theparasiti heatux throughthewi kand theevaporator body. Thepresent

numeri al simulations may improvethe understanding of the physi al me hanisms operating in an LHP

evaporator,andtheir ouplingwiththeother partsoftheLHP,andprovideguidan efortheLHPdesign,

aimingtoredu ethethermalresistan eofthesystem.

Themodel hasbeen onfrontedto asetofexperimental datafrom theliterature. Agoodagreementis

foundbetweenexperimental andtheoreti alresultsfortheentire heatinputrange.

Heattransferthroughthetransportlineshastobetakenintoa ount,in parti ularin variable

ondu -tan emode,sin eit anmodifysigni antlythesuboolingoftheliquidenteringthereservoir.

Simulationsshowthemajorinuen eoftheevaporation oe ientandofthewi k ondu tivityonthe

LHPoperatingtemperatureaswellasonthetemperatureeld in theevaporator. When theee t ofthe

heattransferthroughtheevaporator asingisinsigni ant,the ompetitionbetweentheparasiti heatux

throughthewi kandtheheattransfertotheevaporationzoneleadstoanextremumforwhi htheoperating

temperature is maximal. Additionally, a low evaporation oe ientleads to a signi antin rease of the

(28)

andleadto amajor de reaseoftheevaporationrate. Asaresult,amu h largeroperating temperatureis

foundinthe aseofsubstantialheat ondu tionin theevaporatorbody.

A knowledgements

Theauthors wantto a knowledgethenan ialsupportof theEuropeanCommission throughtheFP7

PRIMAEProje t,Contra tn. 265413(www.primae.org).

Nomen lature

A

ross-se tionalarea [

m

2

℄

a

ev evaporation oe ient

c

p spe i heat [

J.kg

−1

_.K

m.s

−2

℄

H

height [

m

℄

h

onve tiveheattransfer oe ient [

W.m

−2

m

℄

M

molarmass [

kg.mol

−1

℄

˙

m

massowrate [

kg.s

non-dimensionalheattransferrate

R

universalgas onstant [

J.K

−1

_.mol

m.s

−1

℄

x

quality,axis oordinate

y, z

axis oordinates [

m

℄

(29)

ρ

density [

kg.m

−3

℄

λ

thermal ondu tivity [

W.m

−1

_.K

−1

℄

ε

voidfra tion,porosity

µ

dynami vis osity [

P a.s

℄

χ

tt

Martinelliparameter

φ

heatuxdensity [

W.m

−2

℄

(30)

air

amb

ambient

D

diameter e ee tive

eq

equivalent

ev

evaporator,evaporation

f

uid

f ric

fri tional

gr

groove

h

homogeneous

in

inner, inlet

l

liquid

m, n

dis retization step

max

maximum

mom

momentum

out

outer, outlet

par

totalparasiti heatux

res

reservoir

sat

saturation

sen

sensible

static

hydrostati

sub

sub ooling

top

topsurfa e

tot

total

tube

throughthetransportlinewall

v

vapour

w

wi k

wall

evaporator,tubewall

(31)

F r

Froudenumber

Ga

Galileonumber

Gz

Graetznumber

P r

Prandtlnumber

Ra

Rayleighnumber

Re

Reynoldsnumber Referen es

[1℄ J. Ku,Operating hara teristi s of loop heat pipes, in:

29 th

International Conferen e on Environmental System,no.

1999-01-2007,1999.

[2℄ S.Launay, V. Sartre, J. Bonjour, Parametri analysisof loop heat pipeoperation: a literature review, International

JournalofThermalS ien es46(7)(2007)621636.

[3℄ T.Kaya,J.Ku,T.T.Hoang,M.K.Cheung,Mathemati almodelingofloopheatpipes,in:37

th

AIAAAerospa eS ien es

MeetingandExhibit,no.477,1999.

[4℄ A.Adoni,A.Ambirajan,V.Jasvanth,D.Kumar,P.Dutta,Theoreti alstudiesofhardllinginloopheatpipes,Journal

ofThermophysi sandHeatTransfer24(1)(2010)173183.

[5℄ P.Chuang, Animprovedsteady-state model ofloop heatpipesbasedon experimentalandtheoreti al analyses, Ph.D.

thesis,PennsylvaniaStateUniversity,USA(2003).

[6℄ S.Launay,V.Sartre,J.Bonjour,Analyti almodelfor hara terizationofloopheatpipes,JournalofThermophysi sand

HeatTransfer22(4)(2008)623631.

[7℄ L.Bai,G.Lin,H.Zhang,D.Wen,Mathemati almodelingofsteady-stateoperationofaloopheatpipe,AppliedThermal

Engineering29(2009)26432654.

[8℄ Y.Cao,A.Faghri,Analyti alsolutionsofowandheattransferinaporousstru turewithpartialheatingandevaporation

ontheuppersurfa e,InternationalJournalofHeatandMassTransfer37(10)(1994)15251533.

[9℄ C.Figus,Y.LeBray,S.Bories,M.Prat,Heatandmasstransferwithphase hangeinaporousstru turepartiallyheated:

ontinuummodelandporenetworksimulations,InternationalJournalofHeatandMassTransfer42(1999)14461458.

[10℄ T.Coquard, Coupledheatand masstransferinanelementofa apillaryevaporator (infren h), Ph.D.thesis,Institut

NationalPolyte hniquedeToulouse,Fran e(2006).

[11℄ T.Zhao,Q.Liao,On apillary-drivenowandphase- hangeheattransferinaporousstru tureheatedbyannedsurfa e:

measurementsandmodeling,InternationalJournalofHeatandMassTransfer43(7)(2000)11411155.

[12℄ C.Ren,Q.Wu,M.Hu,Heattransferinloopheatpipe'swi k: Ee tofporousstru tureparameters,Journalof

Thermo-physi sandHeatTransfer21(4)(2007)702711.

[13℄ M.A.Chernysheva,Y.F.Maydanik,3D-modelforheatandmasstransfersimulationinatevaporator of opper-water

loopheatpipe,AppliedThermalEngineering33-34(2012)124134.

[14℄ N.Rivière,V.Sartre,J.Bonjour,Fluidmassdistributioninaloopheatpipewithatevaporator,in:

15 th

International

HeatPipeConferen e,no.4-1,Clemson,USA,2010.

[15℄ H.Müller-Steinhagen,K.He k, Asimplefri tionpressuredrop orrelationfortwo-phaseowinpipes,Chemi al

Engi-neeringandPro essing: Pro essIntensi ation20(6)(1986)297308.

[16℄ J.R.Thome,EngineeringDataBookIII,WolverineTubeIn .,WebBook,2006.

[17℄ S.W.Chur hill, H.H.Chu,Correlating equationsforlaminarandturbulentfree onve tionfromahorizontal ylinder,

(32)

[19℄ M. Mis evi ,P.Lavieille,B.Piaud, Numeri alstudyof onve tiveow with ondensation ofa pure uidin apillary

regime,InternationalJournalofHeatandMassTransfer52(2122)(2009)51305140.

[20℄ H.M.Soliman,Ontheannular-to-wavyowpatterntransitionduring ondensationinsidehorizontaltubes,TheCanadian

JournalofChemi alEngineering60(4)(1982)475481.

[21℄ J.G.Collier,J.R.Thome,Conve tiveboilingand ondensation,

3 rd

edition,OxfordUniversityPress,Oxford,UK,1994.

[22℄ M.K.Dobson,J.C.Chato,Condensationinsmoothhorizontaltubes,JournalofHeatTransfer120(1998)193.

[23℄ W.W.Akers, H. A.Deans, O.K.Crosser, Condensingheat transfer within horizontaltubes, Chemi halEngineering

ProgressSymposiumSeries59(29)(1958)171176.

[24℄ M.A.Hanlon, H.B.Ma,Evaporationheattransfer insinteredporousmedia,JournalofHeat Transfer125(4)(2003)

644652.

[25℄ R.Singh,A.Akbarzadeh,M.Mo hizuki,Ee tofwi k hara teristi sonthethermalperforman eoftheminiatureloop

heatpipe,JournalofHeatTransfer131(8)(2009)082601.

[26℄ J.Choi,Y.Lee,B.Sung,C.Kim,Investigationonoperational hara teristi softheminiatureloopheatpipeswithat

evaporatorsbasedondiversevaporremoval hannels,in:

16 th

InternationalHeatPipeConferen e,no.023,Lyon,Fran e,

2012.

[27℄ V.P.Carey,Liquid-vaporphase- hangephenomena,Hemisphere,NewYork,USA,1992.

[28℄ I. Eames,N. Marr, H. Sabir, Theevaporation oe ient ofwater: a review, International Journalof Heat and Mass

Transfer40(12)(1997)29632973.

[29℄ R.Singh,A.Akbarzadeh, M.Mo hizuki,Operational hara teristi sofaminiatureloopheatpipewithatevaporator,