Literature review: Steady-state modelling of loop heat pipes

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Literature review: Steady-state modelling of loop heat pipes

B. Siedel, V. Sartre, Frédéric Lefèvre

To cite this version:

B. Siedel, V. Sartre, Frédéric Lefèvre. Literature review: Steady-state modelling of loop heat pipes.

Applied Thermal Engineering, Elsevier, 2015, 75, pp.709-723. �10.1016/j.applthermaleng.2014.10.030�.

�hal-01286776�

Benjamin Siedel,ValérieSartre

∗

,FrédériLefèvre

UniversitédeLyon,CNRS

INSA-Lyon,CETHILUMR5008,F-69621,Villeurbanne,Frane

UniversitéLyon1,F-69622,Villeurbanne,Frane

Abstrat

Loopheatpipes(LHPs) areeientheattransfersystemswhoseoperationisbasedontheliquid-vapourphase-hange

phenomenon. They use the apillary pressure generated in the porous struture to irulate the uid from the heat

soureto theheatsink. Inthispaper,anexhaustiveliteraturereviewisarriedoutinorderto investigatetheexisting

steady-state models ofLHPs. These modelsanbedivided intothree ategories: the ompletenumerial models, the

numerial evaporatormodels and theanalytialmodels. The mostused models aredesribedand ompared. Finally,

asynthesissummarizesallthesteady-statemodelsfrom theliteraturein aomprehensivetable. Thereviewshowsthe

evolutionofthemodellingworksinthepast15yearsandhighlightstheinreasingdevelopmentof3Dinvestigations.

Keywords: Loop heatpipe, Model,Review, Steady-state,Evaporator

1. Introdution

Loop Heat Pipes (LHPs) are eient heat transfer

devies based on the liquid-vapour phase-hange phe-

nomenon. Theyprovide apassiveheat transferbetween

a heat soure and a heat sink, using the apillary pres-

suretoirulatetheuid. Comparedtoonventionalheat

pipes,LHPsoerseveraladvantagesintermsofexibility,

operationagainstgravityandheattransportapability.

Sinetheirrstsuessfulappliationsintheaerospae

industry, LHPshave gained amajor interestin aeronau-

tisand terrestrialappliations. As aonsequene,many

experimental works have been published to provide use-

fuldatatounderstandthephysialmehanismsgoverning

thesesystemsinvariousoperatingonditions(againstthe

gravity, yogeni appliations, start-up behaviour, et.)

and to optimisetheir design(hoie ofthe working uid,

material of the wik, geometry of the evaporator, et.).

Atthesametime,manytheoretialstudieshavebeenun-

dertakento preditaurately thebehaviourofLHPs,in

partiulartheoupledphenomenaourringintheevapo-

rator/reservoirstruture.

SeveralliteraturereviewsonLHPsarealreadyavailable.

Ku[1℄presentsanextensiveanalysisoftheoperatinghar-

ateristis of loop heat pipes. After explaining theoper-

ating priniples and the thermohydraulis of LHPs, the

authors investigate the LHP behaviour (operating tem-

perature, temperature ontrol,start-up,hystereses,shut-

down)andtheeetoftheevaporatormass,theelevation,

thenon-ondensable gasesand theheatlosses totheam-

∗

Correspondingauthor

Emailaddress: valerie.sartreinsa-lyon .fr (ValérieSartre)

bientontheLHPoperation. SeveralLHPdesignsarealso

disussed.

Maydanik[2℄alsopresentsareviewofdevelopments,re-

sultsoftheoretialanalysesandtestsofLHPs. Thepaper

mainly deals with LHP designs and appliations. Vari-

oustypesofLHPs(large,ontrollable,ramied,reversible,

miniature)areomparedandtheLHPsforbothspaeraft

appliationsandeletronisoolingarepresented.

An extension of these works is given by Launay et al.

[3℄. Theauthors presentanexhaustivereviewof the pa-

rameters aeting the LHP steady-state operation. An

extensiveanalysis of theoperating limitsofLHPs isalso

provided.

AreviewfromAmbirajanetal.[4℄isalsoavailableinthe

literature. After explaining the fundamental oneptsof

theLHP behaviour,the authors disussthe onstrution

details,theoperatingpriniplesandthetypialoperating

harateristis of LHPs. The paper also present urrent

developmentsinmodellingofthermohydraulisanddesign

methodologies. The review of the modelling studies is,

however,farfromexhaustiveandneedsafurtheranalysis.

LaunayandVallée[5℄presentsanexhaustiveoverviewof

theexperimentalstudiespublishedbetween1998and2010.

This review provides a database of experimental results

andhighlightssomeomissionsinthepublishedworksthat

makethedatadiultto useforfurther studies.

Reently, Maydanik et al.[6℄ presenteda literaturere-

viewof developmentsand tests of LHPswith atevapo-

ratordesigns. Theauthorsdisussthevariousgeometrial

ongurations(disk-shaped,retangular,at-oval)andthe

working uids that may beused in eah ase. Then, the

modellingworksonatevaporatorsarepresentedandthe

appliationsofsuhsystemsaredisussed.

WangandYang[7℄arryoutareviewonloopheatpipes

dediated to use in solar water heating. After analysing

theworkingpriniplesofLHPsanddisussingtheexisting

experimental and theoretial works, the authors further

investigatetheopportunities ofusing solarwaterheating

systemswithLHPs.

No exhaustive review on LHP steady-state modelling

studies exists in theliterature. Thispaperintends to in-

trodueaomprehensivereviewoftheexistingtheoretial

worksonthissubjetthathavebeenpublishedsine1999.

Thisworkshouldhelptogiveaglobalviewoftheexisting

modelsintheliteratureandtopointouttheirsimilarities

anddierenes. Italsohighlightsthephysialmehanisms

involved in LHPs that are today still not appropriately

takenintoaountinmostoftheinvestigations.

Most of the theoretial models an be divided into

three ategories, orresponding to omplete numerial

LHPmodels,topreiseevaporatordesriptionsandtoan-

alytialapproahestodesribeLHPs.

2. CompletenumerialLHP models

The majority of omplete numerial LHP models are

based ona volume element disretisationoron eletrial

analogies and desribe the whole devie as a nodal net-

work. The links between the nodes are represented by

thermal resistanes or ondutanesand the energy bal-

ane equationisapplied toeahnode.

Kayaet al.[8℄ developamathematialmodelof aloop

heat pipebasedonthesteady-stateenergybalaneequa-

tionsateahomponentofthesystem. Aylindrialevap-

orator isonsidered. Thefollowingmain assumptionsare

used inthedevelopmentofthemodel:

•

^{The heat transfer through the wik is direted only}

towardstheradialdiretion.

•

^The ompensation hamberand the evaporatorore ontainbothliquidandvapourphases.

•

^TheLHPreahessteady-stateforagivenloopondi- tion.

The total heat load to be dissipated

Q

in

is equal to the

sum of the heat rejeted in the two-phase portion of the

ondenser(latentheat)

Q

^{,theparasitiheatleak}

Q

^hland

theheatlossesfrom thevapourlineto theambient

Q

^vl-a:

Q

ⁱⁿ

= Q

+ Q

^hl

+ Q

^{vl-a (1)}

Intheevaporator,theheatleakompensatesthesubool-

ing of the returning liquid

Q

^{s and the heat losses from}

theompensationhambertotheambient

Q

^-a:

Q

^hl

= Q

^s

+ Q

^{-a (2)}

Toalulate theheat leak,theauthorsonlyonsider on-

dutionthroughthewik,whih anbewritten as:

Q

^hl

= 2πλ

^e

L

^w

ln (D

^w,o

/D

^w,i

) ∆T

^{a,w (3)}

where

λ

^{eistheeetivethermalondutivityofthewik,}

L

^{witslengthand}

D

^w,iand

D

^{w,oitsinnerandouterdiame-}

ters,respetively. Thetemperaturearossthewik

∆T

^a,w

isthedierenebetweentheloalsaturationtemperatures

ausedbythetotalsystempressuredrops

∆P

total

,exlud-

ingthepressuredropinthewikstruture

∆P

w :

∆T

a,w

= ∂T

∂P

sat

(∆P

total

− ∆P

w

)

⁽⁴⁾

Theslopeofthevapour-pressureurve

(∂T /∂P )

^satanbe

alulatedusingtheClausius-Clapeyronrelation. Theto-

tal pressuredrops in the system onsist of the fritional

steady-state pressuredrops in the vapour line

∆P

^{vl, the}

liquidline

∆P

^{ll,theondenser}

∆P

^{,apotentialsubooler}

∆P

^s,thebayonet

∆P

^{bay, theporousstruture}

∆P

^{w and}

thevapourgrooves

∆P

^{vgr. Ifthe LHPis notin horizon-}

talorientation,thepressurediereneassoiatedwiththe

gravityeets

∆P

grav

alsoneedstobetakenintoaount:

∆P

^total

=∆P

^vl

+ ∆P

^ll

+ ∆P

+ ∆P

^s

+ ∆P

^bay

+ ∆P

^w

+ ∆P

^vgr

+ ∆P

^{grav (5)}

Theauthorsemploysingle-phaseorrelationstoalulate

allthefritionalpressuredropsandtakeintoaountthe

owregime(laminarorturbulent)inthealulation. The

relevantpropertiesoftheuidarealulatedwithrespet

to thesaturationtemperature

T

^{sat. Twodistint orrela-}

tionsareusedtoestimatetheeetivethermalondutiv-

ity ofthewik. Todetermineheat lossesto theambient,

theauthorstesteitheranaturalonvetionhypothesisor

aradiativehypothesis.

Thetwo-phaseheatremovalintheondenseronsistsof

twoparts: heatrejetion tothe sinkand heatlossto the

ambient. Thelengthof thetwo-phaseowportionin the

ondenser

L

^,2

ϕ

^{isthengivenby:}

L

^,2

ϕ =Q

Z x

out

x

in

dx [(U A/L)

^,s

(T

^sat

− T

^sink

) +(U A/L)

^,a

(T

^sat

− T

^amb

)] ^{− 1}

⁽⁶⁾

where

(U A/L)

^{,s and}

(U A/L)

^{,a are thethermalondu-}

taneperunit lengthfromthesurfaeoftheondenserto

theheatsinkandtotheambient,respetivelyand

x

^isthe

thermodynamiqualityoftheow:

x

ⁱⁿ

= 1

^and

x

^out

= 0

ifthe total two-phase regionis loatedin theondenser.

T

^{sink and}

T

^{amb arethe}temperaturesofthe heatsinkand theambient, respetively. Theliquid temperature at the

exit of the ondenser is alulated by integration of the

loalheatbalaneonanelementarylengthofthetube

dz

^:

˙ mc

^p,l

dT

dz = (U A/L)

^,a

(T − T

^amb

) + (U A/L)

^,l

(T − T

^sink

)

(7)

wherethemassowrate

m ˙

^is:

˙ m = Q

h

^{lv (8)}

and

c

^p,land

h

^{lv arethespeiheatoftheliquidandthe}

enthalpyofvaporisationoftheuid,respetively.

Thesamemethodisappliedforthepotentialsubooler

and for the liquid line. The subooling of the returning

liquidisthusgivenby:

Q

^s

= ˙ mc

^p,l

(T

^sat

− T

^l,out

)

⁽⁹⁾

where

T

^{l,out istheuid}temperatureat theendoftheliq- uid line. TheLHP operating temperature

T

^{sat is then a}

funtion of

Q

^in,

T

^{sink and}

T

^{amb. An iterativeproedureis}

implementeduntilonvergeneisreahed.

Theauthorsomparetheirmodelto experimentaldata

obtained with the GLAS LHP 1

and another loop devel-

opedfortheNavalResearhLaboratory(NRL). Thepre-

ditions,fortwodistintheatsinktemperatures,arevery

lose to the experimental results. At low powers, some

disrepanies exist showinganeedofamorepreise on-

siderationoftheheatlossesto theambient.

The onsideration of radial mass ow in the wik was

added in a later paper, and new orrelations for natural

onvetionwerealsoomparedtobettertakeintoaount

heatlossestotheambient[10℄. Theauthorsstatethatthe

mathematial modelling ofthe LHPperformanehara-

teristisbeomesmorediultasthesizeoftheLHPde-

reases. Indeed, the lowmass ow rates assoiated with

thelowpowerlevelsinsmallLHPsinduealongerdwelling

time for theworking uid in thetransport lines, despite

smallertubediameters. Therefore,theheatexhangewith

the surroundings beomes moreimportant. Additionally,

heat andmasstransferin smalldiameter tubesis lessin-

vestigated and thus morediult to predit. Aording

to theauthors,thedierenesbetweenthemeasuredand

thealulatedLHPoperatingtemperaturesaremainlyat-

tributed to the inability to predit the overall eetive

thermalondutanearossthewik.

The 1-D steady-state model of Chuang [11℄ is based

ontheenergybalaneequation, thermodynamirelation-

ships and detailed heat transfer and pressure drop al-

ulations in the liquid, vapourand ondenser lines. The

modelinludesthepressuredropsinduedbythebendsin

boththetransportlinesandtheondenser,theonvetive

heattransferbetweentheuidandthewallinthevapour

grooveand bothaxialand radialheat uxes in thewik.

Inahorizontalonguration,pressuredropsinthevapour

line, in theondenser,andthroughthewikare foundto

beinthesameorderofmagnitude,exeptatlowheatloads

for whih the pressuredrops due to the uid ow in the

porous struture are dominant. Moreover, heat transfer

in thevapourhannelsinduesaslightsuperheat(several

Kelvins)attheentraneofthevapourline. Thisstudyde-

sribesextensivelythe LHPoperationin gravity-assisted

onditions. When theLHP isoperatedat apositive ele-

vation (ondenserloatedabovetheevaporator/reservoir,

1

GLASLHP:GeosieneLaserAltimeterSystemPrototypeLoop

Heat Pipe, purhasedbytheNASAin1997 fromDynathermCor-

poration[9℄

evaporator/reservoir in horizontal onguration), it an

operatein apillary-ontrolledmodeorgravity-ontrolled

mode. When theoperation isontrolledby theapillary

fores,thevapourgroovesareonlylled withvapourand

thetotal massowrate

m ˙

total

andiretly bealulated

usingtheheatuxdissipatedbyevaporation

Q

evap :

˙

m

total

= ˙ m

v

= Q

evap

h

^{lv (10)}

where

m ˙

^{v isthevapourmassowratein thegrooves. In}

thegravity-ontrolledmode,theuid owin thehannel

beomestwo-phase. Themassowrateisthereforealu-

latedby:

˙

m

total

= ˙ m

v

+ ˙ m

l

= Q

^evap

h

^lv

+ ˙ m

l

(11)

where

m ˙

^{ldenotestheliquidmassowrateforedintothe}

vapourgroove. Additionally, thepressuregain hasto be

takenintoaountinthepressurebalaneequation. Inthe

gravity-ontrolledmode,thepressuregainfromtheliquid

head ompensates the system total pressure drops. A-

ordingtotheauthors' results,whentheheatloadislow,

theLHPoperatesingravity-ontrolledmodeandthetotal

massowrate in thesystemdoes nothange muhwith

theheatinput. Theoperationisthensimilartothat ofa

thermosyphon. However, for higher heat loads, evapora-

tionarossthemenisiattheoutersurfaeoftheprimary

wiktakesplaeandprovidestheadditionalpressuregain

requiredby the system (apillary-ontrolled mode). The

elevation hasagreat impatontheLHPoperationwhen

theheatloadislowandmodiessigniantlytheshapeof

theharateristiurve. Despiteinterestingresults,heat

transferin the evaporator and the reservoirare not pre-

iselydesribedandtheauthorsstresstheneedofabetter

radial heat leak model based on heat load, temperature

distribution in the primary wik, orientation, properties

oftheprimarywikandvapourqualityin theevaporator

ore.

Adonietal.[12℄developedamathematialmodeltopre-

ditthermalandhydrauliperformaneofanLHP,based

on onservation of mass and energy in the system. The

presentedmodelisvalidforseveralgeometries(ylindrial

orat-plate evaporator, LHPorCPL).Thesamegeneral

method as the one previously desribed is implemented.

Additionally, the pressure drops aross the wik are al-

ulatedusingtheDaryantheoryandspeiorrelations

from the works of El Hajal et al. [13℄ and Thome et al.

[14℄aswellastheFriedelorrelation[15℄areused toal-

ulate the two-phase heat transfer and pressuredrops in

theondenser.

Their model inludes the onsideration of hard-lling,

orrespondingto areservoirfull ofliquid,thusnotallow-

ing a two-phase saturation equilibrium in this loop ele-

ment. In that ase, the energy balane equation in the

LHP ore (Equation2) is solved simultaneously for the

reservoirtemperatureandtheliquid density in thereser-

voir. Indeed, the reservoir temperature determines both

theliquiddensityinsidethereservoir. Thus,knowingthe

exat mass of working uid in the system, it is possible

to determineareservoirtemperaturethatprovidesatthe

sametimeenoughsuboolingandatwo-phaselengththat

isonsistentwithaoodedreservoir.

Theauthorsonludethathard-llingleadstoanearly

xed ondutanemode, an inreaseof heat leaks and a

redutionoftheondensationlengthduetotheexpansion

oftheliquid.

Alaterstudyaddstheonsiderationofthebayonetand

presentstheeetofthemassofworkinguidontheLHP

performane [16℄. To take into aount the presene of

the bayonet, the authors addseveralnodes to the model

(Figure1).

Intheevaporatorore,thereisaheatbalanebetween

the radial heat leak

Q

^{hl, the heat from the ore to the}

reservoir

Q

,r

andthesuboolingofthereturningliquid:

Q

hl

= ˙ mc

p,54e

(T

5

− T

4e

) + Q

,r

(12)

where

c

p,54e

isthemeanspei heatoftheworkinguid

betweentheoutletofthebayonet

(5)

^{andtheinletofthe}

oreinthebayonet

(4e)

^.

Q

^,risequalto:

Q

^,r

= λ

^,r

(T

⁵

− T

^r

)

⁽¹³⁾

where

λ

^{,r is the thermal ondutane between the ore}

and thereservoir. Inthereservoir,theenergybalaneis:

Q

^r,

^∞ = Q

^b,r

+ Q

^,r

+ Q

^{evap,r (14)}

where

Q

^r,

^∞

^{is the heat loss to the ambient,}

Q

^{b,r is the}

heat exhange between the uid and the bayonet in the

reservoirand

Q

^{evap,r is theaxialparasiti heatux. The}

authors desribe two distint states of the reservoir. In-

deed, thethermalouplingbetweenthereservoirandthe

ore stronglydepends onthe volume of working uid in-

side the loop. If the height of liquid in the reservoir is

suh that vapouran exist in the reservoirtubeand the

ore,thenthereservoirhasagoodthermalandhydrauli

ouplingwiththeore. Otherwise,abadthermallinkex-

ists. Whenabayonetispresentintheore,asoldliquid

exits from it, the temperature of the reservoiris higher

thanthatoftheorewhihissubooled. Inaseofagood

thermallink,thevapourgeneratedinthereservoirtravels

to theolderorewhereitondenses.

The authors also further studied the hard-lling phe-

nomena andthe inueneof the bayoneton ahard-lled

reservoir[17℄. Their resultsshow that themass ofwork-

ing uid and the bayonet have a signiant inuene on

theLHPoperation. With largermasses,the heatloadat

whihhardllingoursredues,thusinduingasteeprise

in theoperatingtemperature. Whenabayonetexistsand

the ambienttemperature is higher than that of the heat

sink,thesuperheatofthereservoirmayleadtoadeprime

of the loop. Indeed, in suh aase, thehard-lling leads

to asteep rise of the liquid temperature in the reservoir

enough,boilinginipieneouldourandindueamajor

degradationoreventhefailureoftheloopoperation.

Bai et al. [18℄ also model an LHP (with ylindrial

evaporator) based on energy onservation laws. Their

work shows the inuene of a two-layer ompound wik

(Figure2) andtakesinto onsiderationthe liquid-vapour

shear stressesin theondenser, basedonanannularow

regime and onsidering both phasesindependently. Heat

transferin the evaporator is modelled using anodal net-

work and applying anenergy balaneat eah node. The

various thermalondutaneare estimatedby theexper-

imental data oralulated using aradial 1D approxima-

tionof heat and mass transferin the wik. The desrip-

tionofthetwo-phaseregion inthe ondenser(andin the

transportlinesifthetwo-phasezoneexeedstheondenser

boundaries)isobtainedbynitedierenesolutions. The

vapourqualityand thepressuredropsare thus obtained.

The transport lines are divided into several nodes, eah

of whih representing a ertain ontrol volume, and the

alulationsareonduted at eah node. However,longi-

tudinalondution inthetransport linesisnegletedand

thethermalondutanebetweentheevaporatorwalland

theliquid-vapourinterfaeissetinaordanewithexper-

imentalresults. Thestudyalsodesribesthebehaviourof

aloop with aooded reservoir(hard-lling). Under this

situation,thevolumeexpansionoftheworkinguidinthe

reservoirresultsinthellingofaninreasinglengthofthe

ondenser when the applied heat load and the operating

temperature inrease. The onlusions are the same as

in the work of Adoni et al. [12℄. The authors also on-

dutedaparametrianalysisofaryogeniLHPbasedon

thesamemodel,withtheadditionofseondaryondenser,

evaporatorandompensationhamber[19℄.

Figure2: Shematiof the ross-setion of a two-layer ompound

wik[18℄

Singhetal.[20℄presentasteady-statemodelofanLHP

with a at disk-shaped evaporator on the basis of mass

andenergyonservationpriniplesforseveralontrolvol-

ondenser 4

r

4e

Q

^{load 5}

1

2 3

Figure1:ShematiofanLHPwithaylindrialevaporator andabayonet

umes. Thedesriptionoftheevaporatortakesintoaount

heat lossestotheambientaswellasparasitiheat trans-

fer (Figure3). The total heat load is dissipated by the

evaporation, heat losses to theambient andthe subool-

ing ofthereturningliquid. Single-phaseoworrelations

are used in theondenser andheat losses tothe ambient

are negleted. A ondenser model is developed to take

into aountthen-and-tubegeometry of theondenser.

A global desriptionis presented, dening anoverallsur-

faeeienyofthenarrayandalogmeantemperature

dierene from the n surfae to the ambient air. The

surfaetemperatureofthenarrayisonsidereduniform

and equalto the temperatureof theondenser tube. An

aeptableagreementwiththeexperimentaldataisfound

with anikelwik. However, arather large dierene is

observed between the alulated performane for a op-

per wikand theexperimental results. Aording to the

authors, this is probably due to the shortoming of the

model to onsider aurately theheat and mass transfer

inside theevaporationzone.

˙

m

^l

c

^p,l

(T

^s

− T

^s

)

Q

^,a

Q

^ap

Q

^e,

m ˙

^l

h

^l

˙ m

^v

h

^v

˙

m

^v

(h

^v

− h

^l

)

Figure3:Energybalaneontheevaporator

Rivière et al. [21℄ present aomplete numerial model

of LHP in order to study the inuene of the uid mass

distribution in a loopwith aat evaporator. Themodel

is basedon alassi nodal network forthe onsideration

ofheattransferintheevaporator/reservoir.However,the

vapourgrooves,thetransportlinesandtheondenserare

disretised into small elements and two energy balane

equations are applied on eah element, one for the solid

wallandonefortheuid. Suhadistintionbetweenthe

wall and the uid temperaturesis the main original fea-

tureof this model (Figure4). It enablesto takeinto a-

ountthetemperaturevariationinthevapourgroovesand

in thevapourline, aswell asthelongitudinalondution

throughthetransportline walls. Furthermore,thepossi-

bleourreneofuid ondensationinthevapourlineas

wellasavapourdesuperheatingzoneintheondenserare

onsidered. Theauthors showthat vapourstartsto on-

densate in thevapour line, due to heat lossesto the am-

bient. Theyalsoinvestigatetheinueneofthetransport

linewallthermalondutivityandtheuidmassdistribu-

tionin theLHPduring operation. This modelis further

developed in Siedel et al. [22℄. Theauthors ombine the

monodimensionaldisretisationofthetransportlineswith

a2Ddevelopmentoftheheatandmasstransferinthewik

andintheevaporatorasing,intheaseofadisk-shaped

geometry. Suh an improvementenablesan aurate de-

terminationoftheparasitiheatuxesandtheonsidera-

tionofanaommodationoeientto harateriseheat

transferin theevaporationzone.

Hodotet al.[23℄ developed aglobal LHPmodel, om-

bining ane three-dimensionaldesription of the evapo-

rator/reservoirand amonodimensional thermo-hydrauli

model of the transport lines and theondenser. The 3D

heattransferequationissolvedusingtheOpenFOAM soft-

wareand resultsare presentedforaylindrialgeometry.

Figure4:FluidandwalltemperaturesalongtheLHP[21℄

Convetiveheat transferis taken into aount inside the

reservoir,inthegrooves,aswellasintheporousmedium.

The1Dnodalmodelofthetransportlinesisbasedonthe

work of Rivière et al. [21℄, enabling the onsideration of

the longitudinal ondution in the transport lines. The

authors use thesimulations tooptimise the saddleshape

design(Figure5)andthevapourgroovesnumberandlo-

ation. Suhaompletemodelassoiatesathoroughmod-

ellingofheatandmasstransferin thetransportlinesand

the ondenser with a ne 3D thermal desription of the

evaporator/reservoir,thus enablinganaurate onsider-

ation oftheparasitiheatlossesduring operation.

Figure5: Optimisationofthesaddleshape[23℄

Severalother globalnumerialsteady-statemodels an

be foundin theliterature[2430℄. Theyare summarised

in setion5.

3. Numerial evaporator models

ManynumerialLHPmodelsanbefoundin theliter-

atureandareusefultoolsforthedesignandoptimization

of LHPsaswellasforabetterunderstandingof theou-

pledphenomenainvolvedintheLHPoperation. However,

these models are limited and their major restrition lies

in an inaurate modelling of the phenomena ourring

in the evaporator/reservoir. Indeed, heat transfer inside

the evaporator aswellasheat lossesto theambient have

adeisiveinuene ontheloopoperation,partiularly at

theomponentsonstitutingtheevaporatorreservoir(the

evaporatorasing, thewik, thereservoirwall,the liquid

poolinsidethereservoirandthevapourgrooves)mustbe

evaluatedaurately. Theseheatuxesdependonnumer-

ousparameters: groovedesign, eetivethermal ondu-

tivityofthewik,evaporationheattransfer,thermalon-

dutivity of theevaporator envelopematerial, thermohy-

draulipropertiesoftheuid,et. Asaonsequene,thor-

oughstudieshavebeenundertakentomodelheattransfer

in the evaporating region, in the wik, or in the entire

evaporator/reservoir.

Several theoretial analyses speially investigate the

development of avapour zone inside the porous medium

[3136℄. These studies, based on ontinuum models or

pore-networksimulations,fousonheatandmasstransfer

insidethewikinordertoevaluatethesizeandtheshape

of a potential liquid-vapour interfae inside the porous

struture. Otherinvestigationsassumeaporousstruture

thatisfullysaturatedwithliquid.

3.1. Fully saturatedwik

A majorityof the LHPmodels from the literature as-

sumeaompleteliquidsaturationofthewikandastati

liquid-vapour interfaeat the surfaeof the wik in on-

tatwith thegroove. Inthat ase,themenisi providing

the apillary pressure are all loated in thepores at the

surfaeoftheporousstrutureinontatwiththegrooves

andthewikisfullofliquid.

Li and Peterson [37℄ developed a three-dimensional

steady-statemodelofasquareatevaporatorwithafully

saturatedwik struture. The omputationaldomain in-

ludestheliquid bulkofthereservoir,thewik,agroove

andametalli substratewhere theheatinput isimposed

(Figure6). The 3Dgoverningequationsfor theheatand

mass transfer (ontinuity, Dary and energy) are devel-

oped. A temperature boundary ondition is adopted at

the liquid-vapour interfae, assuming a perfet evapora-

tion rate. Furthermore, no thermal resistane is taken

intoaountfortheontatbetweentheenvelopeandthe

wik. In order to expedite the onvergene of the al-

ulations, a line-by-line iteration and a Tridiagonal Ma-

trixAlgorithmalongwithaThomasalgorithmsolver,and

suessiveunder-relaxationiterativemethods are usedto

obtainedthe three-dimensionaltemperature distribution.

Thetemperatureandpressuredistributionsinthewikare

disussedandtheveloityeldisinvestigated. Thehighest

heat uxours in thewik-n-grooveorner, onrming

the resultsof Demidov and Yatsenko[38℄. Furthermore,

theresultsshowthatthetemperaturediereneisnotsig-

niant along the axial diretion of the groove. Thus, a

two-dimensionalassumptionisaeptableinthemodelling

oftheevaporator.

Zhang et al. [39℄ also developed a 3D model of a at

evaporatorofanLHP.However,inthatase,thereservoir

isadjaentto thewik. Thus,theliquidowenteringthe

wikisperpendiulartotheheatuximposedattheevap-

orator wall. Theomputational domainis approximately

thesameasintheworkofLiandPeterson[37℄(Figure7).

The wik is onsidered to be fully saturated with liquid.

Theuidowinthewikandinthegroovearedetermined

based on theequations of ontinuity, energy, momentum

and Dary. Heat ondutionis alsoomputedin thewall

region. No heat losses to the ambient nor from the wall

to thereservoirare onsidered. Theboundaryonditions

for the wik region are the reservoirtemperature on one

sideand thesaturationtemperatureofthevapourgroove

at the liquid-vapour interfae. The thermal ontat be-

tweenthewallandthewikisonsideredperfet. Anite

volumemethodisintroduedtosolvetheproblem.

The ow and temperature elds in the wik and the

strutural optimisation of the evaporator (loation and

shape of the grooves) are disussed (Figure8). The re-

sultsshowthat thetemperatureat thetopofthewallin-

reasessmoothlyintheaxialdiretionofthegroove. Due

to the eet of evaporation,the temperatureis higherin

the wik than at the interfae betweenthe wik and the

vapourgroove. Theliquidowingthroughthewikissu-

perheatedbeforereahingtheevaporationzone. Thepres-

suredropinduedbytheowinthewikisonlyof

129 Pa

when the heat load is equal to

80 W

⁽

10 W·cm ^{− 2}

^{). An}

investigationisalsomadeabouttheloationandthegeo-

metrial harateristisofthevapourgrooves. Twotypes

ofevaporatorsareompared: onewiththevapourgrooves

mahined inside the wik (Figure8b) and another with

thegroovesinside thewall(Figure8). Whenthegroove

isloatedinsidethewall,theevaporatinginterfaeisonly

loatedatthebottomofthevapourgroove,whihresults

in largertemperaturegradientsin thewik, ahigher su-

perheat of theliquid inside the apillary struture and a

higher temperature of the evaporator heating wall. The

Figure7:Numerialdomainandoordinatesystem[39℄

authors also onlude that the best results are ahieved

with square grooves(ratio height-width equal to

1

^{) and}

withawidth ration-grooverangingfrom

0.5

^to

1

^.

Chernysheva and Maydanik [40℄ present a 3D mathe-

matial model of a omplete at LHP evaporator with

the reservoir adjaent to the porous struture. All the

main strutural elements of the evaporator/reservoirare

inludedinthemodel: body,wik,vapourgrooves,barrier

layerand ompensation hamber (Figure9). The three-

dimensionalheatequationissolvedfortheentireevapora-

tor. Theauthorsonsideranite evaporationheattrans-

fer, thermal ontatresistanebetween thewik and the

bodyandthedryingofthewiksurfae,basedonthenu-

leationtheory. Ifthe loalliquid superheatin thepores

islargerthanaalulatednuleationsuperheat, thewik

surfaeisonsidereddryandnoevaporationoursatthis

partiular spot. Due to theagitation ensuredby theliq-

uid that arrivesfrom the liquid line, the uid inside the

ompensationhamberisonsideredatauniformtemper-

ature. A nite dierene method is omputed to solve

numeriallytheproblem. Themodeladequatelydesribes

thermalproessesintheevaporatorandthespei har-

aterofaone-sidedheatloadsupply. Theauthorsobtain

the3D temperature eld in theentire evaporatoraswell

astheveloityeldsinthegrooves(Figure9). Theresults

showanonuniformityoftheevaporationrateintheentire

ativezone. Indeed,there are low-evaporationzones ow-

ingtotheinsuientheatingoftheperipheralsetionsof

(b) Temperature eld with

groovesinthewik

() Temperature eld with

groovesinthewall

Figure8: Temperatureeldintheatevaporator[39℄

theevaporator. Thevapourgroovesloatedin theentre

of the ative zone ontribute mainly to the evaporation

proess. However, at high heat uxes, large superheats

and a potential drying of the wik may lead to a larger

ativation of the evaporation proesses in the peripheral

setions. About

90 %

^{of thetotal heat loadis dissipated}

throughevaporation.

Chernysheva and Maydanik [41℄ further disuss the

temperature distribution in the evaporator and dierent

phasesofthewikdryingproessforuniformandonen-

tratedheating,basedonthesamemodel. Auniformheat-

ing means that the whole ative zone is heated whereas

in the ase of onentrated heating, the heater oupies

a small part of the ative zone. Another paper presents

alulationsfortheheatandmasstransferintheompen-

sation hamberof the sameevaporator andthe intensity

ofinternalheatexhangeinthereservoirdependingonits

orientation[42℄. Theauthorsmodelheatandmasstrans-

fer proessesin the entireevaporator/reservoirusing the

softwareEFD.Lab 2

. Theyobtainthetemperatureeldin

2

EFD.Lab: aomputationaluiddynamisformerlydistributed

byNIKAGmbH.Thelatestversion,alledFloEFD TM

,isdistributed

byMentorGraphis

r

Figure9: Temperatureeld with

Q

in

= 400 W

;A-top surfaeof thebody,B-atlevelofhalfthegroovedepth, C-atlevelof half

theevaporatorthikness,D-evaporatorviewfromabove[40℄

theevaporatorandtheveloityeld intheompensation

hamber. The latter is onsidered ompletely lledwith

liquid. A onstant heat transferoeient with the am-

bientisassumed. A onstantmassowrateistakeninto

aountfortheentraneintothebayonetaswellasforthe

interfaebetween thewikand the liquidbulk. Further-

more,thesurfaeofthevapourgroovesissetataonstant

temperature. Theresults show that the inuene of the

gravityissigniantonheatandmasstransferinthereser-

voir. Theloalheat transferoeientin theliquidpool

ofthereservoiranreah

600 W · m ^{− 2} · K ^{− 1}

^losetothewik

athighheatloads(Figure10). Thevalueofthemeanheat

exhangeoeientintheompensationhamberisabout

140 W · m ^{− 2} · K ^{− 1}

^{athighheatux (}

Q

ⁱⁿ

= 500 W

^).

Figure10: Heat transferoeienteldatdierentheatloads: a)

100 W

,b)

300 W

,)

500 W

[42℄

3.2. Liquid-vapour interfaein thewik

Theloationoftheliquid-vapourinterfaein theevap-

oratoranhaveasigniantinueneontheheattransfer

inside thewikand ismostlyof interestwheninvestigat-

ing the deprime of the loop following the drying out of

theporousstruture. Indeed,thegrowthofvapourzones

inside thewik leadsto adierentthermal prolein the

wik,to ahangeof theevaporationinterfaeshapeand,

inaseofapenetrationarosstheentireporousstruture,

to afailureoftheentireloopoperation.

Considering heat and mass transfer and evaporation

senko[38℄theoretiallyinvestigatethegrowthofavapour

zoneinsidetheapillarystruture. Theauthorspostulate

theexisteneofavapourbubblebetweenthewikandthe

n,growinginsizeandeventuallyommuniatingwiththe

groove(Figure11).

(a)

(b)

Figure11: Growthof(a) a"large"vapourzone and (b)a"small"

vapourzoneinsidetheapillarystruture[38℄

ThisphenomenonisfurtherstudiedbyFigusetal.[31℄,

whoalsodevelopaporenetworkmodeltoonsiderapore

size distributioninside theporousstruture. Inthis type

ofmodel,theporespaeismodelledbyanetworkofsites

(pores)andbonds(throats),aspresentedinFigure 12. A

omplementarynetworkisonsideredtotakeintoaount

ondutiveheattransfer.

Figure12: Skethofaporenetworkmodel[31℄

At the beginning of the numerial proedure, the net-

work is saturated with liquid exept the rst series of

bondsunderneaththenwhiharesaturatedwithvapour.

Mass, momentum and energybalane equations for eah

elementofthenetworksenablethealulationofthetem-

peratureandthepressureelds. Ifthepressuredierene

arosstheliquid-vapourinterfaeishigherthanthemax-

imalapillarypressure,thebondassoiatedwiththatdif-

fereneisinvadedbythevapour. Onethenetworkphase

distributionhasbeenupdated,theoverallproedureisre-

peateduntilastationarysolutionisfound.

The authors ompare the standard ontinuum model

(basedonontinuousequations)withtheporenetworkone

Toloatethe liquid-vapourinterfae inside thewik, the

ontinuummodelassumesthewiktobelledwithvapour

ifitstemperatureisgreaterthanthesaturationtempera-

ture. Both methods give similar resultsharaterised by

asmoothvapourzoneunderthen. Whenthewikdoes

not have ahomogeneous porosity, a fratal vapour zone

extension is observed (Figure13b). They obtain vapour

breakthroughforheatuxequaltoabout

20 W · cm ^{− 2}

^{. As}

inthepreviouslyitedwork,theauthorsassumethepres-

ene of an initial vapour zone in the wik, initiating the

vapourinvasion proess.

(a)

(b)

Figure13: Pore-network simulationsofthe vapour frontinsidethe

porouswik[31℄: (a) homogeneousporosity(

φ = 5 kW · m ⁻²

): on- tinuummodel(blak)andpore-networkmodel(white);(b)inhomo-

geneousporosity(

φ = 90 kW · m ⁻²

)

Other modelling works have been more reently pub-

lished on this topi, further developing a pore network

model. Coquard [32℄ improvesthe model of Figus et al.

[31℄ , onsidering onvetion in both the liquid and the

vapourphasesandtakingintoaountthevariationofthe

vapourdensity. Heat transferin thegroovesisalulated

andtheenergybalaneisalsoomputedintheevaporator

wall. Moreover, no symmetryis assumed forthe vapour

region. The author develops adual model: the pressure

and temperatureeldsare alulatedusing homogeneous

equations whereasthe apillarity and hene the loation

oftheinterfaeareonsideredusingtheporenetwork. To

determinetheinipieneofthevapourdevelopmentinside

theporousstruture,theauthorarbitrarilyassumesanu-

leation superheat of

3 K

^{. This assumption also implies}

theexisteneofvapourorgasembryosunderthenthat

failitates the nuleation. Aording to the author, the

preseneofthevapourregioninside thewikhasamajor

inueneontheevaporatoroperation. Itinduesanaddi-

tional thermalresistane, leadingto alarge superheat of

thenandtoaninreaseoftheparasitiheatlosses.

Themodelwasfurther developedbyLouriou [33℄to take

into aounttransientphenomena,whiharenotrelevant

to thisreview'stopi.

Kaya and Goldak [34℄ numerially analyse heat and

mass transferin theporous struture ofaloop heatpipe

usinganiteelementmethod. Theystudytheexisteneof

avapourregioninsidethewiktoassesstheboilinglimit

mirosopiavitiesat thewik-ninterfaeforsmallsu-

perheatvaluesasaresultoftrappedgasintheseavities.

Aordingto theauthors, theboilinginipientsuperheat

valueisdiulttopredit,sineitdependsonseveralpa-

rametersinaomplexmanner. Therefore,theyarbitrarily

assume the inipiene of the vapour zone would our if

a superheat of

4 K

^{of the liquid is reahed. However, if}

theontatbetweenthenandthewikisimprovedand

theworkinguidispuriedtothegreatestpossibleextent,

thuspreventingthepreseneofvapourembryostrappedat

thewik-ninterfae,theboilinginipieneanbedelayed

to higher superheats, at the sameorder of magnitude as

thatforhomogeneousnuleationinapureliquid. Theau-

thorsinvestigatesuhasenario,alulatingthesuperheat

limitusingthelusternuleationtheory. Theirexperimen-

tal results indiate no strong transient eets that ould

bethe expeted onsequeneof an explosiveevaporation

atthewik-ninterfae,evenwhentheapplied heatload

ishigherthatthealulatedboilinglimit. Theyonlude

thatavapourregionmustexistunder thenandprovide

anesapepathforthebubblestothegroove,thusprevent-

ingaash-likevapourexpansion.However,theabseneof

strong transient eets does notneessarily onrms the

partialdryingoftheapillarystruture.

All of these numerial works assume initial lusters of

non-ondensablegasestrapped betweenthewikand the

n. Theselusterswouldenabletheexpansionofavapour

zone in theporous struture, requiringonlyalowsuper-

heat. In thease of a good mehanial ontat between

thewikandtheevaporatorbodyandifthepurityofthe

working uid is high, it an be assumed that no vapour

norgaswould initiallyexist intheporousstruture. The

onditionsof boiling initiationare then givenby the ho-

mogeneousnuleation theory. Insuh aase, theboiling

onditionwouldbeahievedataveryhighsuperheat. As

wasexplainedbyMishkinis andOhterbek[43℄andlater

onrmed by Kayaand Goldak [ 34℄, if the LHPis fabri-

atedandlledwithahighdegreeofarefulness,partiu-

larlyforthedegassingoftheliquidandtheeliminationof

non-ondensablegasesinthesystem,pratiallynoboiling

phenomenonistobeexpetedduringoperation.

3.3. Conlusion

Allthepreviousdesribednumerialanalysesgiveabet-

terunderstandingofthephenomenainvolvedin theevap-

oratorofaloopheatpipe. Parasitiheattransfer,heatex-

hangebetweentheevaporatorwallandthegrooves,heat

andmasstransferinthewik,inthevapourgroovesandin

theompensationhamberaswellastheharaterisation

oftheevaporationzoneareinvestigated. Thesenumerial

studiesshowtheomplexityof heat andmasstransferin

aloopevaporator andareausefultoolforimprovingthe

designandthemanufaturingofLHPs. However,theop-

eratingparametersofthemodel(temperatureoftheliquid

returningtotheondenser,pressuredierenebetweenthe

pled with averysimplied loop model. Therefore, there

isalakofknowledgeonerningtheinueneofthephe-

nomenaourring in theevaporatoron theentiresystem

operation.

4. Analytial studies on LHPs

Following inreasing omputational resoures, the ma-

jorpartofthemodellingeortsfousondevelopingmod-

elsusing various numerialmethods. Few researhworks

presentanalytialmodelsofLHPs,inwhihtheoperating

parameters(temperature,pressuredrops,mass owrate,

et.) an be expliitly determined, without the need of

any numerial sheme. Howeverthe analytial approah

doesnotneessitatelargenumerialresouresandanbe

easilyimplementedinasimplesoftware. Therefore,itan

beapowerfultoolforthedesignandoptimisationofloop

heat pipes.

Aording to Launay et al. [44℄, Maydanik et al. [ 45℄

developedananalytialmodelwithalosed-formsolution

basedonanenergybalanein thereservoirandthepres-

surebalaneintheoverallloop. Theradialparasitiheat

transfer through the ylindrial wik wastaken into a-

ount, but the axial heat ux and the heat losses to the

ambient were negleted. Assuming low heat losses from

the liquid line and a heat load equal to the heat dissi-

pated byevaporation, the following simplied expression

wasgiven:

T

^v

= T

^,o

+ (T

^r

− T

^,o

) D

w,o

D

^w,i

Q

in

c

p,l

2πλ

^e

L

^w

h

^lv

(15)

where

T

^v,

T

^rand

T

^{,o arethe}temperaturesinthevapour grooves,in thereservoirandat theendof theondenser,

respetively.

D

w,o

and

D

w,i

aretheouterandinnerdiam-

etersofthewik,respetivelyand

L

^witslength.

λ

^eisthe

eetivethermalondutivity ofthewik. Inthis losed-

formsolution,preditingthevapourtemperaturerequires

theknowledgeofthetemperaturesattheondenseroutlet

andin thereservoir. Therefore,this expressionannotbe

diretlyusedtoexpresstheLHPthermaloperationbased

onitsgeometrialharateristis.

Cao andFaghri [46℄ presentananalytial work forthe

heat and mass transferin aretangular apillary porous

struturewithpartialheatingandevaporationontheup-

per surfae (Figure14a). This geometry an be diretly

related to theevaporatorof aCPL oran LHP. Basedon

symmetryassumptions,theauthorsusethemethodofsep-

aration ofvariables todetermine solutionsin theform of

Fourierseries. Thesidesoftheomputationaldomainare

onsidered adiabati, the bottom boundary ondition is

aset temperatureand theupperboundaryonditionis a

heatinputononesideandaheatoutputontheotherside.

Therefore, analytialsolutionsfortheliquidpressure,ve-

loityand temperatureelds in theporous struture are

obtained(Figure14).

(a)Modellingdomain

(b)Isothermsintheporouswik

()Veloityvetorsinthewik

Figure14: Analytialheatandmasstransferinthewik[46℄

perleft-handorner,underthen. Hightemperaturegra-

dientsareexpetedneartheupperlimit,whereasthetem-

peratureeld ismoreuniformatthebottom. Conerning

mass transfer, the liquid ows vertially into the porous

strutureandremainsnearlyone-dimensionaluntilreah-

ingthemiddlesetionofthewik.

Furukawa[47℄ presentsadesign-orientedanalytialde-

sription of an LHP. His approah is very original and

aimsatoptimisingthedesignoftheLHPingivenoperat-

ingonditions. Theinitialhypothesisistheknowledgeofa

design-speiedoperatingtemperature. Theauthorsolves

the heat and mass transfer equations in the ylindrial

evaporator. Pressure lossesin the loopand heat transfer

intheondenserarealulatedasafuntionofthegeomet-

rialpropertiesofthesystem. Severalperformaneindies

are dened (number of transferunits, temperatureee-

tiveness,ritial Bondnumber,pump eieny) in order

toimprovethedesignoftheLHP.Basedontheoperating

temperatureandonthegeometrialandthermohydrauli

harateristisoftheloop,allthedesignparameters(wik

thikness,transportlines diameter,wikporeradius and

porosity, reservoir volume, ondenser length) are evalu-

ated. The paper presents several harts to optimise the

design harateristisof the LHP. This study is a useful

toolinthesizingoftheLHPomponentsbasedontemper-

ature onstraints. However,in manyases,theoperating

temperatureisnot neessarilytheoperatinglimitand is,

assuh, notapriori known.

Chernyshevaet al.[24℄ present ananalytialinvestiga-

tionoftwoompensationhamberoperatingmodes,either

thehard-llingorthetwo-phasestate. Basedonthether-

modynami relationshipbetween theliquid-vapour inter-

faesinthegrooveandintheondenserorinthereservoir,

theauthorsdevelopananalytialexpressionoftheoperat-

ingtemperature

T

^{ev. Inaseof}hard-lling,theevaporator temperatureisequalto:

T

^ev

=T

^sink

+

1

α

^ond,ext

S

^ond,ext

+ R

^{ond,body (16)}

+ 1

α

^ond,int

S

^ond,int

+ X

i

W

ⁱ

F n

ⁱ

+ 1 α

^ev

S

^q

! Q

^load

where

T

^{sink is the heat sink} temperature and

α

^ond,ext,

α

^ond,intand

α

^{evaretheheattransferoeientattheex-}

ternalsideoftheondenser,attheinternalsideoftheon-

denserandin theevaporationzone,respetively.

S

^ond,ext

and

S

^{ond,intorrespondto theexternalandinternal sur-}

faeareasoftheondenser,respetivelyand

S

^qistheevap-

orator surfae area where heat is supplied.

Q

load is the

totalheat loadto bedissipatedby theloop,

R

ond,body is

the thermal resistaneofthe ondenser wall and

W

^{i and}

F n

^{i are the oeients taking into aount the geomet-}

rial andthermophysial parametersin thealulationof

pressuredropsinthevapourline. Intheaseofanexisting

liquid-vapour interfaeinside theompensation hamber,

T

^ev

= T

+ (∆P

^v

+ ∆P

^l

+ ∆P

^g

) dT dP _T

+ Q

^load

α

ev

S

q

(17)

where

T

isthe temperature in theompensation ham-

ber, and

∆P

v ,

∆P

l

and

∆P

g

are the pressure drops in

thevapourline,in theliquidline,anddue tothegravity,

respetively. Despiteprovidingasimpleexpressionof the

evaporatortemperature,thisdevelopmentshowstwomain

limitations. Firstly,intheaseofasaturatedreservoir,the

operating temperature is afuntion of theompensation

hambertemperature, whih is apriori notknown. Se-

ondly, several major assumptions are made: heat losses

totheambient,parasitiheattransfer,two-phasepressure

dropsandheattransferinthetransportlinesarenegleted.

Suhhypothesesmayleadtoalargeinaurayintheop-

erationpredition.

Launayet al.[44℄ developlosed-formsolutionslinking

theLHPoperatingtemperaturetovariousuidproperties

andgeometrialparameters. Basedonanenergybalane

oneahLHPomponentandonthermodynamiequations

(Figure15),thereservoirtemperature

T

^{ranbepredited}

for both the variable and the xed ondutane modes

(Equation18andEquation19). Intheseexpressions,

K

^C,

K

^suband

K

^{LareglobalondutanesdenedinFigure 15}

and

R

^E,

R

^{w and}

R

^{wall are evaporatorresistanes dened}

inthesamegure.

L

^Land

L

^{Carethelengthsoftheliquid}

line and ofthe ondenser, respetively, and

D

^{L and}

D

^C,i

their respetive diameters.

R

^{A is the thermal resistane}

thatrepresentsheatlossesofthereservoirto theambient

at temperature

T

^A. Additionally, simple analytial solu- tionsoftheheat loadorrespondingto thetransitionbe-

tweenbothmodesareexpressed. Theeetofthegeomet-

rial parameters and uid thermophysial properties on

theLHP operationare learly highlighted. However, the

identiation of the evaporator thermal resistane needs

to be adjusted to experimental data or may require an

additionalevaporatorauratemodel.

ThismodelisfurtherdevelopedbySiedeletal.[48℄. The

heat transfer equation in the evaporator/reservoirstru-

tureis solvedusing aFourierseries development. There-

fore,themodelombinesboththeadvantagesofalosed-

formsolutionwithapreisedeterminationoftheparasiti

heatuxes.

BooandJung[49℄ondut atheoretialmodellingofa

loop heat pipewith a atevaporator. Basedon anodal

network of theevaporator and ofenergy balane at eah

nodeof thesystem,theauthors preditthetemperatures

ofeahomponent. Theproleandthetemperatureofthe

liquid-vapourinterfae in the pores were expressedusing

thethin-lm theory. Theevaporationheattransferoe-

ientisthendependentontheaommodationoeient

andontheheatondutionthroughtheliquidlmtothe

vapour. Transversal heat losses are also taken into a-

ountandaheatexhangerlassialNTUmethodisused

for the modelling of the ondenser. However,no losed-

formsolutionoftheoperatingtemperatureisgivenin the

VCM

T

^r

=

T

^sink

+ h

^lv

c

^p,l

R

^E

R

^wall

+ T

^A

R

^A

Q

ⁱⁿ

+ (T

^A

− T

^sink

)

1 − exp

− πD

^L

L

^L

K

^L

h

^lv

Q

ⁱⁿ

c

^p,l

1 − 1 Q

ⁱⁿ

1 ρ

^v

c

^p,l

1 R

^w

+ 1

R

^wall

(∆P

^v

+ ∆P

^l

− ∆P

^g

) − h

^lv

c

^p,l

R

^A

⁽¹⁸⁾

FCM

T

r

= T

sink

+ Q

ⁱⁿ

πD

^C,i

L

^C

K

^C

1 + R

^E

R

wall

K

^C

K

sub

1 + R

^E

R

^wall

(19)

Figure15: LHPthermalresistanenetwork[44℄

paper. Furthermore,longitudinalparasitiheatlossesare

not onsidered and the reservoir is assumed to be lled

with liquid during operation, whih does notneessarily

orrespondto anatualLHPoperation.

5. Conlusion

The present reviewinvestigated theexisting modelling

studies ofLHPsfrom theliterature. Table1andTable2

summarisethemainsteady-statemodellingworksofLHPs

published in the past years. This survey onsiders both

the omplete models and the partial evaporator models.

However, transientanalyses were omitted, sine theyare

notinthesopeofthepresentstudy.

As presented in this paper, many theoretial works

aboutLHPs have been undertaken in the past15 years.

Mostofthemarenumerialanalyses,basedonnodalnet-

worksoronnitedierenemethods,whereasfewanalyt-

ial studies are developed. Spei odes for LHPs have

been extensively developed in the past years, inluding

morefeaturesand onsideringmoreauratelythephysi-

al phenomena involvedin theloops. However,there are

still onlyfew studiesthat showaomplete desriptionof

the LHP with a preise onsideration of heat and mass

transferin the evaporator/reservoirstruture, despite its

majorsignianeontheloopoperation. Thisonlusion

is aonsequeneof theomplexity ofthe phenomenao-

urringin loopheatpipes.

This summary also shows the development of three-

dimensionalmodelsinthereentyears,followingtheavail-

abilityoflargeromputationalresoures. Flatevaporators

havealsobeenmoreinvestigatedinthelastyearsandshow

thegainofinterestforthisgeometryofevaporator,ason-

rmedby Maydanik et al. [6℄. The partial drying of the

wik and the hard-llingare phenomena that have been

seldomonsidered. However,mostof the modelsinvesti-

gate intensively heat and mass transfer in the transport

linesandtheondenser.

Asexplainedinthispaper,theliteraturepresentsanex-

tensivenumberofsteady-statemodels. Thesemodelsare

usefultoolstopreditthethermalperformaneofanLHP,

to understand the oupled physial mehanisms involved

in these systems,to estimate theinuene of various pa-

rametersonthe behaviourof LHPsandto improvetheir

design. Thisdiversityprovidesalargetheoretialdatabase

fortheommunityinvestigatingloopheat pipes.

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