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Literature review: Steady-state modelling of loop heatpipes

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�

Literature review: Steady-state modelling of loop heat pipesBenjamin Siedel, Valérie Sartre∗, Frédéri LefèvreUniversité de Lyon, CNRSINSA-Lyon, CETHIL UMR5008, F-69621, Villeurbanne, Fran eUniversité Lyon 1, F-69622, Villeurbanne, Fran eAbstra tLoop heat pipes (LHPs) are e� ient heat transfer systems whose operation is based on the liquid-vapour phase- hangephenomenon. They use the apillary pressure generated in the porous stru ture to ir ulate the �uid from the heatsour e to the heat sink. In this paper, an exhaustive literature review is arried out in order to investigate the existingsteady-state models of LHPs. These models an be divided into three ategories: the omplete numeri al models, thenumeri al evaporator models and the analyti al models. The most used models are des ribed and ompared. Finally,a synthesis summarizes all the steady-state models from the literature in a omprehensive table. The review shows theevolution of the modelling works in the past 15 years and highlights the in reasing development of 3D investigations.Keywords: Loop heat pipe, Model, Review, Steady-state, Evaporator1. Introdu tionLoop Heat Pipes (LHPs) are e� ient heat transferdevi es based on the liquid-vapour phase- hange phe-nomenon. They provide a passive heat transfer betweena heat sour e and a heat sink, using the apillary pres-sure to ir ulate the �uid. Compared to onventional heatpipes, LHPs o�er several advantages in terms of �exibility,operation against gravity and heat transport apability.Sin e their �rst su essful appli ations in the aerospa eindustry, LHPs have gained a major interest in aeronau-ti s and terrestrial appli ations. As a onsequen e, manyexperimental works have been published to provide use-ful data to understand the physi al me hanisms governingthese systems in various operating onditions (against thegravity, yogeni appli ations, start-up behaviour, et .)and to optimise their design ( hoi e of the working �uid,material of the wi k, geometry of the evaporator, et .).At the same time, many theoreti al studies have been un-dertaken to predi t a urately the behaviour of LHPs, inparti ular the oupled phenomena o urring in the evapo-rator/reservoir stru ture.Several literature reviews on LHPs are already available.Ku [1℄ presents an extensive analysis of the operating har-a teristi s of loop heat pipes. After explaining the oper-ating prin iples and the thermohydrauli s of LHPs, theauthors investigate the LHP behaviour (operating tem-perature, temperature ontrol, start-up, hystereses, shut-down) and the e�e t of the evaporator mass, the elevation,the non- ondensable gases and the heat losses to the am-

∗Corresponding authorEmail address: valerie.sartre�insa-lyon.fr (Valérie Sartre)

bient on the LHP operation. Several LHP designs are alsodis ussed.Maydanik [2℄ also presents a review of developments, re-sults of theoreti al analyses and tests of LHPs. The papermainly deals with LHP designs and appli ations. Vari-ous types of LHPs (large, ontrollable, rami�ed, reversible,miniature) are ompared and the LHPs for both spa e raftappli ations and ele troni s ooling are presented.An extension of these works is given by Launay et al.[3℄. The authors present an exhaustive review of the pa-rameters a�e ting the LHP steady-state operation. Anextensive analysis of the operating limits of LHPs is alsoprovided.A review from Ambirajan et al. [4℄ is also available in theliterature. After explaining the fundamental on epts ofthe LHP behaviour, the authors dis uss the onstru tiondetails, the operating prin iples and the typi al operating hara teristi s of LHPs. The paper also present urrentdevelopments in modelling of thermohydrauli s and designmethodologies. The review of the modelling studies is,however, far from exhaustive and needs a further analysis.Launay and Vallée [5℄ presents an exhaustive overview ofthe experimental studies published between 1998 and 2010.This review provides a database of experimental resultsand highlights some omissions in the published works thatmake the data di� ult to use for further studies.Re ently, Maydanik et al. [6℄ presented a literature re-view of developments and tests of LHPs with �at evapo-rator designs. The authors dis uss the various geometri al on�gurations (disk-shaped, re tangular, �at-oval) and theworking �uids that may be used in ea h ase. Then, themodelling works on �at evaporators are presented and theappli ations of su h systems are dis ussed.Preprint submitted to Applied Thermal Engineering July 22, 2014

Wang and Yang [7℄ arry out a review on loop heat pipesdedi ated to use in solar water heating. After analysingthe working prin iples of LHPs and dis ussing the existingexperimental and theoreti al works, the authors furtherinvestigate the opportunities of using solar water heatingsystems with LHPs.No exhaustive review on LHP steady-state modellingstudies exists in the literature. This paper intends to in-trodu e a omprehensive review of the existing theoreti alworks on this subje t that have been published sin e 1999.This work should help to give a global view of the existingmodels in the literature and to point out their similaritiesand di�eren es. It also highlights the physi al me hanismsinvolved in LHPs that are today still not appropriatelytaken into a ount in most of the investigations.Most of the theoreti al models an be divided intothree ategories, orresponding to omplete numeri alLHP models, to pre ise evaporator des riptions and to an-alyti al approa hes to des ribe LHPs.2. Complete numeri al LHP modelsThe majority of omplete numeri al LHP models arebased on a volume element dis retisation or on ele tri alanalogies and des ribe the whole devi e as a nodal net-work. The links between the nodes are represented bythermal resistan es or ondu tan es and the energy bal-an e equation is applied to ea h node.Kaya et al. [8℄ develop a mathemati al model of a loopheat pipe based on the steady-state energy balan e equa-tions at ea h omponent of the system. A ylindri al evap-orator is onsidered. The following main assumptions areused in the development of the model:• The heat transfer through the wi k is dire ted onlytowards the radial dire tion.• The ompensation hamber and the evaporator ore ontain both liquid and vapour phases.• The LHP rea hes steady-state for a given loop ondi-tion.The total heat load to be dissipated Qin is equal to thesum of the heat reje ted in the two-phase portion of the ondenser (latent heat) Q , the parasiti heat leak Qhl andthe heat losses from the vapour line to the ambient Qvl-a:

Qin = Q +Qhl +Qvl-a (1)In the evaporator, the heat leak ompensates the sub ool-ing of the returning liquid Qs and the heat losses fromthe ompensation hamber to the ambient Q -a:Qhl = Qs +Q -a (2)To al ulate the heat leak, the authors only onsider on-du tion through the wi k, whi h an be written as:

Qhl = 2πλe�Lwln (Dw,o/Dw,i)∆Ta ,w (3)

where λe� is the e�e tive thermal ondu tivity of the wi k,Lw its length and Dw,i and Dw,o its inner and outer diame-ters, respe tively. The temperature a ross the wi k ∆Ta ,wis the di�eren e between the lo al saturation temperatures aused by the total system pressure drops ∆Ptotal, ex lud-ing the pressure drop in the wi k stru ture ∆Pw:

∆Ta ,w =

(

∂T

∂P

)sat (∆Ptotal −∆Pw) (4)The slope of the vapour-pressure urve (∂T/∂P )sat an be al ulated using the Clausius-Clapeyron relation. The to-tal pressure drops in the system onsist of the fri tionalsteady-state pressure drops in the vapour line ∆Pvl, theliquid line ∆Pll, the ondenser ∆P , a potential sub ooler∆Ps , the bayonet ∆Pbay, the porous stru ture ∆Pw andthe vapour grooves ∆Pvgr. If the LHP is not in horizon-tal orientation, the pressure di�eren e asso iated with thegravity e�e ts ∆Pgrav also needs to be taken into a ount:∆Ptotal =∆Pvl +∆Pll +∆P +∆Ps +∆Pbay +∆Pw

+∆Pvgr +∆Pgrav (5)The authors employ single-phase orrelations to al ulateall the fri tional pressure drops and take into a ount the�ow regime (laminar or turbulent) in the al ulation. Therelevant properties of the �uid are al ulated with respe tto the saturation temperature Tsat. Two distin t orrela-tions are used to estimate the e�e tive thermal ondu tiv-ity of the wi k. To determine heat losses to the ambient,the authors test either a natural onve tion hypothesis ora radiative hypothesis.The two-phase heat removal in the ondenser onsists oftwo parts: heat reje tion to the sink and heat loss to theambient. The length of the two-phase �ow portion in the ondenser L ,2ϕ is then given by:L ,2ϕ =Q ∫ xout

xin dx [(UA/L) ,s (Tsat − Tsink)+(UA/L) ,a (Tsat − Tamb)]−1 (6)where (UA/L) ,s and (UA/L) ,a are the thermal ondu -tan e per unit length from the surfa e of the ondenser tothe heat sink and to the ambient, respe tively and x is thethermodynami quality of the �ow: xin = 1 and xout = 0if the total two-phase region is lo ated in the ondenser.

Tsink and Tamb are the temperatures of the heat sink andthe ambient, respe tively. The liquid temperature at theexit of the ondenser is al ulated by integration of thelo al heat balan e on an elementary length of the tube dz:mcp,ldT

dz= (UA/L) ,a (T − Tamb) + (UA/L) ,l (T − Tsink)(7)where the mass �ow rate m is:

m =Q hlv (8)2

and cp,l and hlv are the spe i� heat of the liquid and theenthalpy of vaporisation of the �uid, respe tively.The same method is applied for the potential sub oolerand for the liquid line. The sub ooling of the returningliquid is thus given by:Qs = mcp,l (Tsat − Tl,out) (9)where Tl,out is the �uid temperature at the end of the liq-uid line. The LHP operating temperature Tsat is then afun tion of Qin, Tsink and Tamb. An iterative pro edure isimplemented until onvergen e is rea hed.The authors ompare their model to experimental dataobtained with the GLAS LHP1 and another loop devel-oped for the Naval Resear h Laboratory (NRL). The pre-di tions, for two distin t heat sink temperatures, are very lose to the experimental results. At low powers, somedis repan ies exist showing a need of a more pre ise on-sideration of the heat losses to the ambient.The onsideration of radial mass �ow in the wi k wasadded in a later paper, and new orrelations for natural onve tion were also ompared to better take into a ountheat losses to the ambient [10℄. The authors state that themathemati al modelling of the LHP performan e hara -teristi s be omes more di� ult as the size of the LHP de- reases. Indeed, the low mass �ow rates asso iated withthe low power levels in small LHPs indu e a longer dwellingtime for the working �uid in the transport lines, despitesmaller tube diameters. Therefore, the heat ex hange withthe surroundings be omes more important. Additionally,heat and mass transfer in small diameter tubes is less in-vestigated and thus more di� ult to predi t. A ordingto the authors, the di�eren es between the measured andthe al ulated LHP operating temperatures are mainly at-tributed to the inability to predi t the overall e�e tivethermal ondu tan e a ross the wi k.The 1-D steady-state model of Chuang [11℄ is basedon the energy balan e equation, thermodynami relation-ships and detailed heat transfer and pressure drop al- ulations in the liquid, vapour and ondenser lines. Themodel in ludes the pressure drops indu ed by the bends inboth the transport lines and the ondenser, the onve tiveheat transfer between the �uid and the wall in the vapourgroove and both axial and radial heat �uxes in the wi k.In a horizontal on�guration, pressure drops in the vapourline, in the ondenser, and through the wi k are found tobe in the same order of magnitude, ex ept at low heat loadsfor whi h the pressure drops due to the �uid �ow in theporous stru ture are dominant. Moreover, heat transferin the vapour hannels indu es a slight superheat (severalKelvins) at the entran e of the vapour line. This study de-s ribes extensively the LHP operation in gravity-assisted onditions. When the LHP is operated at a positive ele-vation ( ondenser lo ated above the evaporator/reservoir,1GLAS LHP: Geos ien e Laser Altimeter System Prototype LoopHeat Pipe, pur hased by the NASA in 1997 from Dynatherm Cor-poration [9℄

evaporator/reservoir in horizontal on�guration), it anoperate in apillary- ontrolled mode or gravity- ontrolledmode. When the operation is ontrolled by the apillaryfor es, the vapour grooves are only �lled with vapour andthe total mass �ow rate mtotal an dire tly be al ulatedusing the heat �ux dissipated by evaporation Qevap:mtotal = mv =

Qevaphlv (10)where mv is the vapour mass �ow rate in the grooves. Inthe gravity- ontrolled mode, the �uid �ow in the hannelbe omes two-phase. The mass �ow rate is therefore al u-lated by:

mtotal = mv + ml = Qevaphlv + ml (11)where ml denotes the liquid mass �ow rate for ed into thevapour groove. Additionally, the pressure gain has to betaken into a ount in the pressure balan e equation. In thegravity- ontrolled mode, the pressure gain from the liquidhead ompensates the system total pressure drops. A - ording to the authors' results, when the heat load is low,the LHP operates in gravity- ontrolled mode and the totalmass �ow rate in the system does not hange mu h withthe heat input. The operation is then similar to that of athermosyphon. However, for higher heat loads, evapora-tion a ross the menis i at the outer surfa e of the primarywi k takes pla e and provides the additional pressure gainrequired by the system ( apillary- ontrolled mode). Theelevation has a great impa t on the LHP operation whenthe heat load is low and modi�es signi� antly the shape ofthe hara teristi urve. Despite interesting results, heattransfer in the evaporator and the reservoir are not pre- isely des ribed and the authors stress the need of a betterradial heat leak model based on heat load, temperaturedistribution in the primary wi k, orientation, propertiesof the primary wi k and vapour quality in the evaporator ore.Adoni et al. [12℄ developed a mathemati al model to pre-di t thermal and hydrauli performan e of an LHP, basedon onservation of mass and energy in the system. Thepresented model is valid for several geometries ( ylindri alor �at-plate evaporator, LHP or CPL). The same generalmethod as the one previously des ribed is implemented.Additionally, the pressure drops a ross the wi k are al- ulated using the Dar yan theory and spe i� orrelationsfrom the works of El Hajal et al. [13℄ and Thome et al.[14℄ as well as the Friedel orrelation [15℄ are used to al- ulate the two-phase heat transfer and pressure drops inthe ondenser.Their model in ludes the onsideration of hard-�lling, orresponding to a reservoir full of liquid, thus not allow-ing a two-phase saturation equilibrium in this loop ele-ment. In that ase, the energy balan e equation in theLHP ore (Equation 2) is solved simultaneously for thereservoir temperature and the liquid density in the reser-voir. Indeed, the reservoir temperature determines both3

the sub ooling of the liquid entering the ondenser andthe liquid density inside the reservoir. Thus, knowing theexa t mass of working �uid in the system, it is possibleto determine a reservoir temperature that provides at thesame time enough sub ooling and a two-phase length thatis onsistent with a �ooded reservoir.The authors on lude that hard-�lling leads to an early�xed ondu tan e mode, an in rease of heat leaks and aredu tion of the ondensation length due to the expansionof the liquid.A later study adds the onsideration of the bayonet andpresents the e�e t of the mass of working �uid on the LHPperforman e [16℄. To take into a ount the presen e ofthe bayonet, the authors add several nodes to the model(Figure 1).In the evaporator ore, there is a heat balan e betweenthe radial heat leak Qhl, the heat from the ore to thereservoir Q ,r and the sub ooling of the returning liquid:Qhl = mcp,54e(T5 − T4e) +Q ,r (12)where cp,54e is the mean spe i� heat of the working �uidbetween the outlet of the bayonet (5) and the inlet of the ore in the bayonet (4e). Q ,r is equal to:

Q ,r = λ ,r(T5 − Tr) (13)where λ ,r is the thermal ondu tan e between the oreand the reservoir. In the reservoir, the energy balan e is:Qr,∞ = Qb,r +Q ,r +Qevap,r (14)where Qr,∞ is the heat loss to the ambient, Qb,r is theheat ex hange between the �uid and the bayonet in thereservoir and Qevap,r is the axial parasiti heat �ux. Theauthors des ribe two distin t states of the reservoir. In-deed, the thermal oupling between the reservoir and the ore strongly depends on the volume of working �uid in-side the loop. If the height of liquid in the reservoir issu h that vapour an exist in the reservoir tube and the ore, then the reservoir has a good thermal and hydrauli oupling with the ore. Otherwise, a bad thermal link ex-ists. When a bayonet is present in the ore, as old liquidexits from it, the temperature of the reservoir is higherthan that of the ore whi h is sub ooled. In ase of a goodthermal link, the vapour generated in the reservoir travelsto the older ore where it ondenses.The authors also further studied the hard-�lling phe-nomena and the in�uen e of the bayonet on a hard-�lledreservoir [17℄. Their results show that the mass of work-ing �uid and the bayonet have a signi� ant in�uen e onthe LHP operation. With larger masses, the heat load atwhi h hard �lling o urs redu es, thus indu ing a steep risein the operating temperature. When a bayonet exists andthe ambient temperature is higher than that of the heatsink, the superheat of the reservoir may lead to a deprimeof the loop. Indeed, in su h a ase, the hard-�lling leadsto a steep rise of the liquid temperature in the reservoir

ompared to the ore temperature. If the superheat is highenough, boiling in ipien e ould o ur and indu e a majordegradation or even the failure of the loop operation.Bai et al. [18℄ also model an LHP (with ylindri alevaporator) based on energy onservation laws. Theirwork shows the in�uen e of a two-layer ompound wi k(Figure 2) and takes into onsideration the liquid-vapourshear stresses in the ondenser, based on an annular �owregime and onsidering both phases independently. Heattransfer in the evaporator is modelled using a nodal net-work and applying an energy balan e at ea h node. Thevarious thermal ondu tan e are estimated by the exper-imental data or al ulated using a radial 1D approxima-tion of heat and mass transfer in the wi k. The des rip-tion of the two-phase region in the ondenser (and in thetransport lines if the two-phase zone ex eeds the ondenserboundaries) is obtained by �nite di�eren e solutions. Thevapour quality and the pressure drops are thus obtained.The transport lines are divided into several nodes, ea hof whi h representing a ertain ontrol volume, and the al ulations are ondu ted at ea h node. However, longi-tudinal ondu tion in the transport lines is negle ted andthe thermal ondu tan e between the evaporator wall andthe liquid-vapour interfa e is set in a ordan e with exper-imental results. The study also des ribes the behaviour ofa loop with a �ooded reservoir (hard-�lling). Under thissituation, the volume expansion of the working �uid in thereservoir results in the �lling of an in reasing length of the ondenser when the applied heat load and the operatingtemperature in rease. The on lusions are the same asin the work of Adoni et al. [12℄. The authors also on-du ted a parametri analysis of a ryogeni LHP based onthe same model, with the addition of se ondary ondenser,evaporator and ompensation hamber [19℄.

Figure 2: S hemati of the ross-se tion of a two-layer ompoundwi k [18℄Singh et al. [20℄ present a steady-state model of an LHPwith a �at disk-shaped evaporator on the basis of massand energy onservation prin iples for several ontrol vol-4

ondenserevaporator4 r 4e

Qload 5 123Figure 1: S hemati of an LHP with a ylindri al evaporator and a bayonetumes. The des ription of the evaporator takes into a ountheat losses to the ambient as well as parasiti heat trans-fer (Figure 3). The total heat load is dissipated by theevaporation, heat losses to the ambient and the sub ool-ing of the returning liquid. Single-phase �ow orrelationsare used in the ondenser and heat losses to the ambientare negle ted. A ondenser model is developed to takeinto a ount the �n-and-tube geometry of the ondenser.A global des ription is presented, de�ning an overall sur-fa e e� ien y of the �n array and a log mean temperaturedi�eren e from the �n surfa e to the ambient air. Thesurfa e temperature of the �n array is onsidered uniformand equal to the temperature of the ondenser tube. Ana eptable agreement with the experimental data is foundwith a ni kel wi k. However, a rather large di�eren e isobserved between the al ulated performan e for a op-per wi k and the experimental results. A ording to theauthors, this is probably due to the short oming of themodel to onsider a urately the heat and mass transferinside the evaporation zone.

mlcp,l(Ts − Ts ) Q ,aQap

Qe, mlhlmvhv

mv(hv − hl)Figure 3: Energy balan e on the evaporator

Rivière et al. [21℄ present a omplete numeri al modelof LHP in order to study the in�uen e of the �uid massdistribution in a loop with a �at evaporator. The modelis based on a lassi nodal network for the onsiderationof heat transfer in the evaporator/reservoir. However, thevapour grooves, the transport lines and the ondenser aredis retised into small elements and two energy balan eequations are applied on ea h element, one for the solidwall and one for the �uid. Su h a distin tion between thewall and the �uid temperatures is the main original fea-ture of this model (Figure 4). It enables to take into a - ount the temperature variation in the vapour grooves andin the vapour line, as well as the longitudinal ondu tionthrough the transport line walls. Furthermore, the possi-ble o urren e of �uid ondensation in the vapour line aswell as a vapour desuperheating zone in the ondenser are onsidered. The authors show that vapour starts to on-densate in the vapour line, due to heat losses to the am-bient. They also investigate the in�uen e of the transportline wall thermal ondu tivity and the �uid mass distribu-tion in the LHP during operation. This model is furtherdeveloped in Siedel et al. [22℄. The authors ombine themonodimensional dis retisation of the transport lines witha 2D development of the heat and mass transfer in the wi kand in the evaporator asing, in the ase of a disk-shapedgeometry. Su h an improvement enables an a urate de-termination of the parasiti heat �uxes and the onsidera-tion of an a ommodation oe� ient to hara terise heattransfer in the evaporation zone.Hodot et al. [23℄ developed a global LHP model, om-bining a �ne three-dimensional des ription of the evapo-rator/reservoir and a monodimensional thermo-hydrauli model of the transport lines and the ondenser. The 3Dheat transfer equation is solved using the OpenFOAM soft-ware and results are presented for a ylindri al geometry.5

Figure 4: Fluid and wall temperatures along the LHP [21℄Conve tive heat transfer is taken into a ount inside thereservoir, in the grooves, as well as in the porous medium.The 1D nodal model of the transport lines is based on thework of Rivière et al. [21℄, enabling the onsideration ofthe longitudinal ondu tion in the transport lines. Theauthors use the simulations to optimise the saddle shapedesign (Figure 5) and the vapour grooves number and lo- ation. Su h a omplete model asso iates a thorough mod-elling of heat and mass transfer in the transport lines andthe ondenser with a �ne 3D thermal des ription of theevaporator/reservoir, thus enabling an a urate onsider-ation of the parasiti heat losses during operation.Figure 5: Optimisation of the saddle shape [23℄Several other global numeri al steady-state models anbe found in the literature [24�30℄. They are summarisedin se tion 5.3. Numeri al evaporator modelsMany numeri al LHP models an be found in the liter-ature and are useful tools for the design and optimizationof LHPs as well as for a better understanding of the ou-pled phenomena involved in the LHP operation. However,these models are limited and their major restri tion liesin an ina urate modelling of the phenomena o urringin the evaporator/reservoir. Indeed, heat transfer insidethe evaporator as well as heat losses to the ambient havea de isive in�uen e on the loop operation, parti ularly at

low heat loads. Therefore, heat and mass transfer betweenthe omponents onstituting the evaporator reservoir (theevaporator asing, the wi k, the reservoir wall, the liquidpool inside the reservoir and the vapour grooves) must beevaluated a urately. These heat �uxes depend on numer-ous parameters: groove design, e�e tive thermal ondu -tivity of the wi k, evaporation heat transfer, thermal on-du tivity of the evaporator envelope material, thermohy-drauli properties of the �uid, et . As a onsequen e, thor-ough studies have been undertaken to model heat transferin the evaporating region, in the wi k, or in the entireevaporator/reservoir.Several theoreti al analyses spe i� ally investigate thedevelopment of a vapour zone inside the porous medium[31�36℄. These studies, based on ontinuum models orpore-network simulations, fo us on heat and mass transferinside the wi k in order to evaluate the size and the shapeof a potential liquid-vapour interfa e inside the porousstru ture. Other investigations assume a porous stru turethat is fully saturated with liquid.3.1. Fully saturated wi kA majority of the LHP models from the literature as-sume a omplete liquid saturation of the wi k and a stati liquid-vapour interfa e at the surfa e of the wi k in on-ta t with the groove. In that ase, the menis i providingthe apillary pressure are all lo ated in the pores at thesurfa e of the porous stru ture in onta t with the groovesand the wi k is full of liquid.Li and Peterson [37℄ developed a three-dimensionalsteady-state model of a square �at evaporator with a fullysaturated wi k stru ture. The omputational domain in- ludes the liquid bulk of the reservoir, the wi k, a grooveand a metalli substrate where the heat input is imposed(Figure 6). The 3D governing equations for the heat andmass transfer ( ontinuity, Dar y and energy) are devel-oped. A temperature boundary ondition is adopted atthe liquid-vapour interfa e, assuming a perfe t evapora-tion rate. Furthermore, no thermal resistan e is takeninto a ount for the onta t between the envelope and thewi k. In order to expedite the onvergen e of the al- ulations, a line-by-line iteration and a Tridiagonal Ma-trix Algorithm along with a Thomas algorithm solver, andsu essive under-relaxation iterative methods are used toobtained the three-dimensional temperature distribution.The temperature and pressure distributions in the wi k aredis ussed and the velo ity �eld is investigated. The highestheat �ux o urs in the wi k-�n-groove orner, on�rmingthe results of Demidov and Yatsenko [38℄. Furthermore,the results show that the temperature di�eren e is not sig-ni� ant along the axial dire tion of the groove. Thus, atwo-dimensional assumption is a eptable in the modellingof the evaporator.Zhang et al. [39℄ also developed a 3D model of a �atevaporator of an LHP. However, in that ase, the reservoiris adja ent to the wi k. Thus, the liquid �ow entering the6

Figure 6: View of the omputational domain [37℄wi k is perpendi ular to the heat �ux imposed at the evap-orator wall. The omputational domain is approximatelythe same as in the work of Li and Peterson [37℄ (Figure 7).The wi k is onsidered to be fully saturated with liquid.The �uid �ow in the wi k and in the groove are determinedbased on the equations of ontinuity, energy, momentumand Dar y. Heat ondu tion is also omputed in the wallregion. No heat losses to the ambient nor from the wallto the reservoir are onsidered. The boundary onditionsfor the wi k region are the reservoir temperature on oneside and the saturation temperature of the vapour grooveat the liquid-vapour interfa e. The thermal onta t be-tween the wall and the wi k is onsidered perfe t. A �nitevolume method is introdu ed to solve the problem.The �ow and temperature �elds in the wi k and thestru tural optimisation of the evaporator (lo ation andshape of the grooves) are dis ussed (Figure 8). The re-sults show that the temperature at the top of the wall in- reases smoothly in the axial dire tion of the groove. Dueto the e�e t of evaporation, the temperature is higher inthe wi k than at the interfa e between the wi k and thevapour groove. The liquid �owing through the wi k is su-perheated before rea hing the evaporation zone. The pres-sure drop indu ed by the �ow in the wi k is only of 129Pawhen the heat load is equal to 80W (10W·cm−2). Aninvestigation is also made about the lo ation and the geo-metri al hara teristi s of the vapour grooves. Two typesof evaporators are ompared: one with the vapour groovesma hined inside the wi k (Figure 8b) and another withthe grooves inside the wall (Figure 8 ). When the grooveis lo ated inside the wall, the evaporating interfa e is onlylo ated at the bottom of the vapour groove, whi h resultsin larger temperature gradients in the wi k, a higher su-perheat of the liquid inside the apillary stru ture and ahigher temperature of the evaporator heating wall. The

Figure 7: Numeri al domain and oordinate system [39℄authors also on lude that the best results are a hievedwith square grooves (ratio height-width equal to 1) andwith a width ratio �n-groove ranging from 0.5 to 1.Chernysheva and Maydanik [40℄ present a 3D mathe-mati al model of a omplete �at LHP evaporator withthe reservoir adja ent to the porous stru ture. All themain stru tural elements of the evaporator/reservoir arein luded in the model: body, wi k, vapour grooves, barrierlayer and ompensation hamber (Figure 9). The three-dimensional heat equation is solved for the entire evapora-tor. The authors onsider a �nite evaporation heat trans-fer, thermal onta t resistan e between the wi k and thebody and the drying of the wi k surfa e, based on the nu- leation theory. If the lo al liquid superheat in the poresis larger than a al ulated nu leation superheat, the wi ksurfa e is onsidered dry and no evaporation o urs at thisparti ular spot. Due to the agitation ensured by the liq-uid that arrives from the liquid line, the �uid inside the ompensation hamber is onsidered at a uniform temper-ature. A �nite di�eren e method is omputed to solvenumeri ally the problem. The model adequately des ribesthermal pro esses in the evaporator and the spe i� har-a ter of a one-sided heat load supply. The authors obtainthe 3D temperature �eld in the entire evaporator as wellas the velo ity �elds in the grooves (Figure 9). The resultsshow a nonuniformity of the evaporation rate in the entirea tive zone. Indeed, there are low-evaporation zones ow-ing to the insu� ient heating of the peripheral se tions of7

(a) Temperature ontours along the axial dire tion

(b) Temperature �eld withgrooves in the wi k ( ) Temperature �eld withgrooves in the wallFigure 8: Temperature �eld in the �at evaporator [39℄the evaporator. The vapour grooves lo ated in the entreof the a tive zone ontribute mainly to the evaporationpro ess. However, at high heat �uxes, large superheatsand a potential drying of the wi k may lead to a largera tivation of the evaporation pro esses in the peripheralse tions. About 90% of the total heat load is dissipatedthrough evaporation.Chernysheva and Maydanik [41℄ further dis uss thetemperature distribution in the evaporator and di�erentphases of the wi k drying pro ess for uniform and on en-trated heating, based on the same model. A uniform heat-ing means that the whole a tive zone is heated whereasin the ase of on entrated heating, the heater o upiesa small part of the a tive zone. Another paper presents al ulations for the heat and mass transfer in the ompen-sation hamber of the same evaporator and the intensityof internal heat ex hange in the reservoir depending on itsorientation [42℄. The authors model heat and mass trans-fer pro esses in the entire evaporator/reservoir using thesoftware EFD.Lab2. They obtain the temperature �eld in2EFD.Lab: a omputational �uid dynami s formerly distributedby NIKA GmbH. The latest version, alled FloEFDTM, is distributedby Mentor Graphi sr

Figure 9: Temperature �eld with Qin = 400W; A - top surfa e ofthe body, B - at level of half the groove depth, C - at level of halfthe evaporator thi kness, D - evaporator view from above [40℄the evaporator and the velo ity �eld in the ompensation hamber. The latter is onsidered ompletely �lled withliquid. A onstant heat transfer oe� ient with the am-bient is assumed. A onstant mass �ow rate is taken intoa ount for the entran e into the bayonet as well as for theinterfa e between the wi k and the liquid bulk. Further-more, the surfa e of the vapour grooves is set at a onstanttemperature. The results show that the in�uen e of thegravity is signi� ant on heat and mass transfer in the reser-voir. The lo al heat transfer oe� ient in the liquid poolof the reservoir an rea h 600W·m−2

·K−1 lose to the wi kat high heat loads (Figure 10). The value of the mean heatex hange oe� ient in the ompensation hamber is about140W·m−2

·K−1 at high heat �ux (Qin = 500W).8

Figure 10: Heat transfer oe� ient �eld at di�erent heat loads: a)100W, b) 300W, ) 500W [42℄3.2. Liquid-vapour interfa e in the wi kThe lo ation of the liquid-vapour interfa e in the evap-orator an have a signi� ant in�uen e on the heat transferinside the wi k and is mostly of interest when investigat-ing the deprime of the loop following the drying out ofthe porous stru ture. Indeed, the growth of vapour zonesinside the wi k leads to a di�erent thermal pro�le in thewi k, to a hange of the evaporation interfa e shape and,in ase of a penetration a ross the entire porous stru ture,to a failure of the entire loop operation.Considering heat and mass transfer and evaporation

pro esses in onventional heat pipes, Demidov and Yat-senko [38℄ theoreti ally investigate the growth of a vapourzone inside the apillary stru ture. The authors postulatethe existen e of a vapour bubble between the wi k and the�n, growing in size and eventually ommuni ating with thegroove (Figure 11).(a) (b)Figure 11: Growth of (a) a "large" vapour zone and (b) a "small"vapour zone inside the apillary stru ture [38℄This phenomenon is further studied by Figus et al. [31℄,who also develop a pore network model to onsider a poresize distribution inside the porous stru ture. In this typeof model, the pore spa e is modelled by a network of sites(pores) and bonds (throats), as presented in Figure 12. A omplementary network is onsidered to take into a ount ondu tive heat transfer.

Figure 12: Sket h of a pore network model [31℄At the beginning of the numeri al pro edure, the net-work is saturated with liquid ex ept the �rst series ofbonds underneath the �n whi h are saturated with vapour.Mass, momentum and energy balan e equations for ea helement of the networks enable the al ulation of the tem-perature and the pressure �elds. If the pressure di�eren ea ross the liquid-vapour interfa e is higher than the max-imal apillary pressure, the bond asso iated with that dif-feren e is invaded by the vapour. On e the network phasedistribution has been updated, the overall pro edure is re-peated until a stationary solution is found.The authors ompare the standard ontinuum model(based on ontinuous equations) with the pore network one9

(Figure 13a), for a onstant pore size in the entire wi k.To lo ate the liquid-vapour interfa e inside the wi k, the ontinuum model assumes the wi k to be �lled with vapourif its temperature is greater than the saturation tempera-ture. Both methods give similar results hara terised bya smooth vapour zone under the �n. When the wi k doesnot have a homogeneous porosity, a fra tal vapour zoneextension is observed (Figure 13b). They obtain vapourbreakthrough for heat �ux equal to about 20W·cm−2. Asin the previously ited work, the authors assume the pres-en e of an initial vapour zone in the wi k, initiating thevapour invasion pro ess.(a) (b)Figure 13: Pore-network simulations of the vapour front inside theporous wi k [31℄: (a) homogeneous porosity (φ = 5kW·m−2): on-tinuum model (bla k) and pore-network model (white); (b) inhomo-geneous porosity (φ = 90 kW·m

−2)Other modelling works have been more re ently pub-lished on this topi , further developing a pore networkmodel. Coquard [32℄ improves the model of Figus et al.[31℄, onsidering onve tion in both the liquid and thevapour phases and taking into a ount the variation of thevapour density. Heat transfer in the grooves is al ulatedand the energy balan e is also omputed in the evaporatorwall. Moreover, no symmetry is assumed for the vapourregion. The author develops a dual model: the pressureand temperature �elds are al ulated using homogeneousequations whereas the apillarity and hen e the lo ationof the interfa e are onsidered using the pore network. Todetermine the in ipien e of the vapour development insidethe porous stru ture, the author arbitrarily assumes a nu- leation superheat of 3K. This assumption also impliesthe existen e of vapour or gas embryos under the �n thatfa ilitates the nu leation. A ording to the author, thepresen e of the vapour region inside the wi k has a majorin�uen e on the evaporator operation. It indu es an addi-tional thermal resistan e, leading to a large superheat ofthe �n and to an in rease of the parasiti heat losses.The model was further developed by Louriou [33℄ to takeinto a ount transient phenomena, whi h are not relevantto this review' s topi .Kaya and Goldak [34℄ numeri ally analyse heat andmass transfer in the porous stru ture of a loop heat pipeusing a �nite element method. They study the existen e ofa vapour region inside the wi k to assess the boiling limit

of LHPs. The authors expe t nu leation to start in themi ros opi avities at the wi k-�n interfa e for small su-perheat values as a result of trapped gas in these avities.A ording to the authors, the boiling in ipient superheatvalue is di� ult to predi t, sin e it depends on several pa-rameters in a omplex manner. Therefore, they arbitrarilyassume the in ipien e of the vapour zone would o ur ifa superheat of 4K of the liquid is rea hed. However, ifthe onta t between the �n and the wi k is improved andthe working �uid is puri�ed to the greatest possible extent,thus preventing the presen e of vapour embryos trapped atthe wi k-�n interfa e, the boiling in ipien e an be delayedto higher superheats, at the same order of magnitude asthat for homogeneous nu leation in a pure liquid. The au-thors investigate su h a s enario, al ulating the superheatlimit using the luster nu leation theory. Their experimen-tal results indi ate no strong transient e�e ts that ouldbe the expe ted onsequen e of an explosive evaporationat the wi k-�n interfa e, even when the applied heat loadis higher that the al ulated boiling limit. They on ludethat a vapour region must exist under the �n and providean es ape path for the bubbles to the groove, thus prevent-ing a �ash-like vapour expansion. However, the absen e ofstrong transient e�e ts does not ne essarily on�rms thepartial drying of the apillary stru ture.All of these numeri al works assume initial lusters ofnon- ondensable gases trapped between the wi k and the�n. These lusters would enable the expansion of a vapourzone in the porous stru ture, requiring only a low super-heat. In the ase of a good me hani al onta t betweenthe wi k and the evaporator body and if the purity of theworking �uid is high, it an be assumed that no vapournor gas would initially exist in the porous stru ture. The onditions of boiling initiation are then given by the ho-mogeneous nu leation theory. In su h a ase, the boiling ondition would be a hieved at a very high superheat. Aswas explained by Mishkinis and O hterbe k [43℄ and later on�rmed by Kaya and Goldak [34℄, if the LHP is fabri- ated and �lled with a high degree of arefulness, parti u-larly for the degassing of the liquid and the elimination ofnon- ondensable gases in the system, pra ti ally no boilingphenomenon is to be expe ted during operation.3.3. Con lusionAll the previous des ribed numeri al analyses give a bet-ter understanding of the phenomena involved in the evap-orator of a loop heat pipe. Parasiti heat transfer, heat ex- hange between the evaporator wall and the grooves, heatand mass transfer in the wi k, in the vapour grooves and inthe ompensation hamber as well as the hara terisationof the evaporation zone are investigated. These numeri alstudies show the omplexity of heat and mass transfer ina loop evaporator and are a useful tool for improving thedesign and the manufa turing of LHPs. However, the op-erating parameters of the model (temperature of the liquidreturning to the ondenser, pressure di�eren e between the10

groove and the reservoir) are either arbitrarily set or ou-pled with a very simpli�ed loop model. Therefore, thereis a la k of knowledge on erning the in�uen e of the phe-nomena o urring in the evaporator on the entire systemoperation.4. Analyti al studies on LHPsFollowing in reasing omputational resour es, the ma-jor part of the modelling e�orts fo us on developing mod-els using various numeri al methods. Few resear h workspresent analyti al models of LHPs, in whi h the operatingparameters (temperature, pressure drops, mass �ow rate,et .) an be expli itly determined, without the need ofany numeri al s heme. However the analyti al approa hdoes not ne essitate large numeri al resour es and an beeasily implemented in a simple software. Therefore, it anbe a powerful tool for the design and optimisation of loopheat pipes.A ording to Launay et al. [44℄, Maydanik et al. [45℄developed an analyti al model with a losed-form solutionbased on an energy balan e in the reservoir and the pres-sure balan e in the overall loop. The radial parasiti heattransfer through the ylindri al wi k was taken into a - ount, but the axial heat �ux and the heat losses to theambient were negle ted. Assuming low heat losses fromthe liquid line and a heat load equal to the heat dissi-pated by evaporation, the following simpli�ed expressionwas given:Tv = T ,o + (Tr − T ,o)(Dw,o

Dw,i ) Qincp,l2πλe�Lwhlv (15)where Tv, Tr and T ,o are the temperatures in the vapourgrooves, in the reservoir and at the end of the ondenser,respe tively. Dw,o and Dw,i are the outer and inner diam-eters of the wi k, respe tively and Lw its length. λe� is thee�e tive thermal ondu tivity of the wi k. In this losed-form solution, predi ting the vapour temperature requiresthe knowledge of the temperatures at the ondenser outletand in the reservoir. Therefore, this expression annot bedire tly used to express the LHP thermal operation basedon its geometri al hara teristi s.Cao and Faghri [46℄ present an analyti al work for theheat and mass transfer in a re tangular apillary porousstru ture with partial heating and evaporation on the up-per surfa e (Figure 14a). This geometry an be dire tlyrelated to the evaporator of a CPL or an LHP. Based onsymmetry assumptions, the authors use the method of sep-aration of variables to determine solutions in the form ofFourier series. The sides of the omputational domain are onsidered adiabati , the bottom boundary ondition isa set temperature and the upper boundary ondition is aheat input on one side and a heat output on the other side.Therefore, analyti al solutions for the liquid pressure, ve-lo ity and temperature �elds in the porous stru ture areobtained (Figure 14).

(a) Modelling domain

(b) Isotherms in the porous wi k

( ) Velo ity ve tors in the wi kFigure 14: Analyti al heat and mass transfer in the wi k [46℄11

The results show the maximum temperature at the up-per left-hand orner, under the �n. High temperature gra-dients are expe ted near the upper limit, whereas the tem-perature �eld is more uniform at the bottom. Con erningmass transfer, the liquid �ows verti ally into the porousstru ture and remains nearly one-dimensional until rea h-ing the middle se tion of the wi k.Furukawa [47℄ presents a design-oriented analyti al de-s ription of an LHP. His approa h is very original andaims at optimising the design of the LHP in given operat-ing onditions. The initial hypothesis is the knowledge of adesign-spe i�ed operating temperature. The author solvesthe heat and mass transfer equations in the ylindri alevaporator. Pressure losses in the loop and heat transferin the ondenser are al ulated as a fun tion of the geomet-ri al properties of the system. Several performan e indi esare de�ned (number of transfer units, temperature e�e -tiveness, riti al Bond number, pump e� ien y) in orderto improve the design of the LHP. Based on the operatingtemperature and on the geometri al and thermohydrauli hara teristi s of the loop, all the design parameters (wi kthi kness, transport lines diameter, wi k pore radius andporosity, reservoir volume, ondenser length) are evalu-ated. The paper presents several harts to optimise thedesign hara teristi s of the LHP. This study is a usefultool in the sizing of the LHP omponents based on temper-ature onstraints. However, in many ases, the operatingtemperature is not ne essarily the operating limit and is,as su h, not a priori known.Chernysheva et al. [24℄ present an analyti al investiga-tion of two ompensation hamber operating modes, eitherthe hard-�lling or the two-phase state. Based on the ther-modynami relationship between the liquid-vapour inter-fa es in the groove and in the ondenser or in the reservoir,the authors develop an analyti al expression of the operat-ing temperature Tev. In ase of hard-�lling, the evaporatortemperature is equal to:Tev =Tsink + (

1

α ond,extS ond,ext +R ond,body (16)+

1

α ond,intS ond,int +∑

i

WiFni + 1

αevSq)Qloadwhere Tsink is the heat sink temperature and α ond,ext,α ond,int and αev are the heat transfer oe� ient at the ex-ternal side of the ondenser, at the internal side of the on-denser and in the evaporation zone, respe tively. S ond,extand S ond,int orrespond to the external and internal sur-fa e areas of the ondenser, respe tively and Sq is the evap-orator surfa e area where heat is supplied. Qload is thetotal heat load to be dissipated by the loop, R ond,body isthe thermal resistan e of the ondenser wall and Wi andFni are the oe� ients taking into a ount the geomet-ri al and thermophysi al parameters in the al ulation ofpressure drops in the vapour line. In the ase of an existingliquid-vapour interfa e inside the ompensation hamber,

the equation is modi�ed to:Tev = T + (∆Pv +∆Pl +∆Pg) dT

dP

∣

∣

∣

∣

T

+QloadαevSq (17)where T is the temperature in the ompensation ham-ber, and ∆Pv, ∆Pl and ∆Pg are the pressure drops inthe vapour line, in the liquid line, and due to the gravity,respe tively. Despite providing a simple expression of theevaporator temperature, this development shows two mainlimitations. Firstly, in the ase of a saturated reservoir, theoperating temperature is a fun tion of the ompensation hamber temperature, whi h is a priori not known. Se -ondly, several major assumptions are made: heat lossesto the ambient, parasiti heat transfer, two-phase pressuredrops and heat transfer in the transport lines are negle ted.Su h hypotheses may lead to a large ina ura y in the op-eration predi tion.Launay et al. [44℄ develop losed-form solutions linkingthe LHP operating temperature to various �uid propertiesand geometri al parameters. Based on an energy balan eon ea h LHP omponent and on thermodynami equations(Figure 15), the reservoir temperature Tr an be predi tedfor both the variable and the �xed ondu tan e modes(Equation 18 and Equation 19). In these expressions, KC,

Ksub and KL are global ondu tan es de�ned in Figure 15and RE, Rw and Rwall are evaporator resistan es de�nedin the same �gure. LL and LC are the lengths of the liquidline and of the ondenser, respe tively, and DL and DC,itheir respe tive diameters. RA is the thermal resistan ethat represents heat losses of the reservoir to the ambientat temperature TA. Additionally, simple analyti al solu-tions of the heat load orresponding to the transition be-tween both modes are expressed. The e�e t of the geomet-ri al parameters and �uid thermophysi al properties onthe LHP operation are learly highlighted. However, theidenti� ation of the evaporator thermal resistan e needsto be adjusted to experimental data or may require anadditional evaporator a urate model.This model is further developed by Siedel et al. [48℄. Theheat transfer equation in the evaporator/reservoir stru -ture is solved using a Fourier series development. There-fore, the model ombines both the advantages of a losed-form solution with a pre ise determination of the parasiti heat �uxes.Boo and Jung [49℄ ondu t a theoreti al modelling of aloop heat pipe with a �at evaporator. Based on a nodalnetwork of the evaporator and of energy balan e at ea hnode of the system, the authors predi t the temperaturesof ea h omponent. The pro�le and the temperature of theliquid-vapour interfa e in the pores were expressed usingthe thin-�lm theory. The evaporation heat transfer oe�- ient is then dependent on the a ommodation oe� ientand on the heat ondu tion through the liquid �lm to thevapour. Transversal heat losses are also taken into a - ount and a heat ex hanger lassi al NTU method is usedfor the modelling of the ondenser. However, no losed-form solution of the operating temperature is given in the12

VCM Tr = Tsink + hlvcp,l [ RE

Rwall + TARAQin ]+ (TA − Tsink) [1− exp

(

−

πDLLLKLhlvQincp,l )]

1−1

Qin [

1

ρvcp,l ( 1

Rw +1

Rwall) (∆Pv +∆Pl −∆Pg)− hlvcp,lRA ] (18)FCM Tr = Tsink + Qin

πDC,iLCKC 1 +RERwall KC

Ksub1 +

RERwall (19)

Figure 15: LHP thermal resistan e network [44℄paper. Furthermore, longitudinal parasiti heat losses arenot onsidered and the reservoir is assumed to be �lledwith liquid during operation, whi h does not ne essarily orrespond to an a tual LHP operation.5. Con lusionThe present review investigated the existing modellingstudies of LHPs from the literature. Table 1 and Table 2summarise the main steady-state modelling works of LHPspublished in the past years. This survey onsiders boththe omplete models and the partial evaporator models.However, transient analyses were omitted, sin e they arenot in the s ope of the present study.As presented in this paper, many theoreti al worksabout LHPs have been undertaken in the past 15 years.Most of them are numeri al analyses, based on nodal net-works or on �nite di�eren e methods, whereas few analyt-i al studies are developed. Spe i� odes for LHPs havebeen extensively developed in the past years, in ludingmore features and onsidering more a urately the physi- al phenomena involved in the loops. However, there arestill only few studies that show a omplete des ription ofthe LHP with a pre ise onsideration of heat and mass

transfer in the evaporator/reservoir stru ture, despite itsmajor signi� an e on the loop operation. This on lusionis a onsequen e of the omplexity of the phenomena o - urring in loop heat pipes.This summary also shows the development of three-dimensional models in the re ent years, following the avail-ability of larger omputational resour es. Flat evaporatorshave also been more investigated in the last years and showthe gain of interest for this geometry of evaporator, as on-�rmed by Maydanik et al. [6℄. The partial drying of thewi k and the hard-�lling are phenomena that have beenseldom onsidered. However, most of the models investi-gate intensively heat and mass transfer in the transportlines and the ondenser.As explained in this paper, the literature presents an ex-tensive number of steady-state models. These models areuseful tools to predi t the thermal performan e of an LHP,to understand the oupled physi al me hanisms involvedin these systems, to estimate the in�uen e of various pa-rameters on the behaviour of LHPs and to improve theirdesign. This diversity provides a large theoreti al databasefor the ommunity investigating loop heat pipes.Referen es[1℄ Jentung Ku. Operating hara teristi s of loop heat pipes. In29

th International Conferen e on Environmental System, Den-ver, Colorado (USA), 1999.[2℄ Yu.F. Maydanik. Loop heat pipes. Applied Thermal Engineer-ing, 25:635�657, 2005.[3℄ S. Launay, V. Sartre, and J. Bonjour. Parametri analysis ofloop heat pipe operation: a literature review. InternationalJournal of Thermal S ien es, 46(7):621�636, 2007.[4℄ A. Ambirajan, A.A. Adoni, J.S. Vaidya, A.A. Rajendran,D. Kumar, and P. Dutta. Loop heat pipes: A review of funda-mentals, operation, and design. Heat Transfer Engineering, 33(4-5):387�405, 2012.[5℄ Stéphane Launay and Martial Vallée. State-of-the-art experi-mental studies on loop heat pipes. Frontiers in Heat Pipes, 2(1), 2011.[6℄ Yu. F. Maydanik, M. A. Chernysheva, and V. G. Pastukhov.Review: Loop heat pipes with �at evaporators. Applied Ther-mal Engineering, 2014.[7℄ Zhangyuan Wang and Wansheng Yang. A review on loop heatpipe for use in solar water heating. Energy and Buildings, 79:143�154, August 2014.[8℄ Tarik Kaya, Jentung Ku, Triem T. Hoang, and Mark K. Che-ung. Mathemati al modeling of loop heat pipes. In 37th AIAAAerospa e S ien es Meeting and Exhibit, Reno, Nevada (USA),1999.13

[9℄ D. Douglas, J. Ku, and T. Kaya. Testing of the geos ien e laseraltimeter system (GLAS) prototype loop heat pipe. In 37thAIAA Aerospa e S ien es Meeting and Exhibit, Reno, Nevada(USA), 1999. Goddard Spa e Flight Center, NASA. Paper 99-0473.[10℄ Tarik Kaya and Jentung Ku. Thermal operational hara teris-ti s of a small-loop heat pipe. Journal of Thermophysi s andHeat Transfer, 17(4):464�470, 2003.[11℄ P.Y.A. Chuang. An improved steady-state model of loop heatpipes based on experimental and theoreti al analyses. PhD the-sis, Pennsylvania State University, 2003.[12℄ A.A. Adoni, A. Ambirajan, VS Jasvanth, D. Kumar, P. Dutta,and K. Slinivasan. Thermohydrauli modeling of apillarypumped loop and loop heat pipe. Journal of Thermophysi sand Heat Transfer, 21(2):410�421, 2007.[13℄ J. El Hajal, J. R. Thome, and A. Cavallini. Condensation inhorizontal tubes. part 1: two-phase �ow pattern map. Interna-tional Journal of Heat and Mass Transfer, 46(18):3349�3363,2003.[14℄ J. R. Thome, J. El Hajal, and A. Cavallini. Condensation inhorizontal tubes. part 2: new heat transfer model based on �owregimes. International Journal of Heat and Mass Transfer, 46(18):3365�3387, 2003.[15℄ L. Friedel. Improved fri tion pressure drop orrelations for hori-zontal and verti al two-phase pipe �ow. In European Two-PhaseFlow Group Meeting, volume 2, Ispra, Italy, 1979.[16℄ Abhijit A. Adoni, Amrit Ambirajan, V. S. Jasvanth, DineshKumar, and Pradip Dutta. E�e ts of mass of harge on loopheat pipe operational hara teristi s. Journal of Thermophysi sand Heat Transfer, 23(2):346�355, 2009.[17℄ A.A. Adoni, A. Ambirajan, VS Jasvanth, D. Kumar, andP. Dutta. Theoreti al studies of hard �lling in loop heat pipes.Journal of Thermophysi s and Heat Transfer, 24(1):173�183,2010.[18℄ Lizhan Bai, Guiping Lin, Hongxing Zhang, and DongshengWen. Mathemati al modeling of steady-state operation of aloop heat pipe. Applied Thermal Engineering, 29:2643�2654,2009.[19℄ L. Bai, G. Lin, and D. Wen. Parametri analysis of steady-stateoperation of a CLHP. Applied Thermal Engineering, 30(8-9):850�858, 2010.[20℄ Randeep Singh, Aliakbar Akbarzadeh, Chris Dixon, and Masa-taka Mo hizuki. Theoreti al modelling of miniature loop heatpipe. Heat and Mass Transfer, 46(2):209�224, 2009.[21℄ Nathanaël Rivière, Valérie Sartre, and Jo elyn Bonjour. Fluidmass distribution in a loop heat pipe with �at evaporator. In

15th International Heat Pipe Conferen e, Clemson, South Car-olina (USA), 2010.[22℄ Benjamin Siedel, Valérie Sartre, and Frédéri Lefèvre. Numeri- al investigation of the thermohydrauli behaviour of a ompleteloop heat pipe. Applied Thermal Engineering, 61(2):541�553,2013.[23℄ R. Hodot, V. Sartre, F. Lefèvre, and C. Sarno. 3D modelingand optimization of a loop heat pipe evaporator. In 17th Inter-national Heat Pipe Conferen e, Kanpur, India, 2013.[24℄ M.A. Chernysheva, S.V. Vershinin, and Yu. F. Maydanik. Op-erating temperature and distribution of a working �uid in LHP.International Journal of Heat and Mass Transfer, 50:2704�2713, 2007.[25℄ Brent Cullimore and Jane Baumann. Steady-state and transientloop heat pipe modeling. In 30th International Conferen e onEnvironmental Systems, Toulouse, Fran e, 2000.[26℄ J. Kim and E. Golliher. Steady state model of a mi ro loopheat pipe. In 18

th Annual IEEE Symposium , Semi ondu torThermal Measurement and Management, pages 137�144, SanJose, California (USA), 2002.[27℄ Mohammed Hamdan, Frank M. Gerner, and H. T. Henderson.Steady state model of a loop heat pipe (LHP) with oherentporous sili on (CPS) wi k in the evaporator. In 19th AnnualIEEE Symposium , Semi ondu tor Thermal Measurement andManagement, pages 88�96, San Jose, California (USA), 2003.

[28℄ M. Ghajar, J. Darabi, and N Crews Jr. A hybrid CFD-mathemati al model for simulation of a MEMS loop heat pipefor ele troni s ooling appli ations. Journal of Mi rome hani sand Mi roengineering, 15:313�321, 2005.[29℄ Yun-Ze Li, Yu-Ying Wang, and Kok-Meng Lee. Dynami mod-eling and transient performan e analysis of a LHP-MEMS ther-mal management system for spa e raft ele troni s. IEEE Trans-a tions on Components and Pa kaging Te hnologies, 33(3):597�606, 2010.[30℄ Mohammad Hamdan and Emad Elnajjar. Thermodynami an-alyti al model of a loop heat pipe. Heat and Mass Transfer, 46(2):167�173, 2009.[31℄ C. Figus, Y. Le Bray, S. Bories, and Mar Prat. Heat andmass transfer with phase hange in a porous stru ture partiallyheated: ontinuum model and pore network simulations. In-ternational Journal of Heat and Mass Transfer, 42:1446�1458,1999.[32℄ Typhaine Coquard. Transferts ouplés de masse et de haleurdans un élément d'évaporateur apillaire. PhD thesis, InstitutNational Polyte hnique de Toulouse, 2006.[33℄ Clément Louriou. Modélisation instationnaire des transferts demasse et de haleur au sein des évaporateurs apillaires. PhDthesis, Institut National Polyte hnique de Toulouse, 2010.[34℄ Tarik Kaya and John Goldak. Numeri al analysis of heat andmass transfer in the apillary stru ture of a loop heat pipe. In-ternational Journal of Heat and Mass Transfer, 49:3211�3220,2006.[35℄ T.S Zhao and Q Liao. On apillary-driven �ow and phase- hange heat transfer in a porous stru ture heated by a �nnedsurfa e: measurements and modeling. International Journal ofHeat and Mass Transfer, 43(7):1141�1155, 2000.[36℄ MA Chernysheva and Y.F. Maydanik. Heat and mass transferin evaporator of loop heat pipe. Journal of Thermophysi s andHeat Transfer, 23(4):725�731, 2009.[37℄ J. Li and GP Peterson. 3D heat transfer analysis in a loopheat pipe evaporator with a fully saturated wi k. InternationalJournal of Heat and Mass Transfer, 54(1):564�574, 2011.[38℄ A.S. Demidov and E.S. Yatsenko. Investigation of heat andmass transfer in the evaporation zone of a heat pipe operatingby the ' inverted menis us' prin iple. International Journal ofHeat and Mass Transfer, 37(14):2155�2163, 1994.[39℄ Xianfeng Zhang, Xuanyou Li, and Shuangfeng Wang. Three-dimensional simulation on heat transfer in the �at evaporatorof miniature loop heat pipe. International Journal of ThermalS ien es, 54:188�198, 2012.[40℄ Mariya A. Chernysheva and Yury F. Maydanik. 3D-model forheat and mass transfer simulation in �at evaporator of opper-water loop heat pipe. Applied Thermal Engineering, 33-34:124�134, 2012.[41℄ Mariya A. Chernysheva and Yury F. Maydanik. Simulationof thermal pro esses in a �at evaporator of a opper�waterloop heat pipe under uniform and on entrated heating. Inter-national Journal of Heat and Mass Transfer, 55(25�26):7385�7397, 2012.[42℄ Mariya A. Chernysheva, Vladimir G. Pastukhov, and Yury F.Maydanik. Analysis of heat ex hange in the ompensation hamber of a loop heat pipe. Energy, 55:253�262, 2013.[43℄ D. Mishkinis and JM O hterbe k. Homogeneous nu leation andthe heat-pipe boiling limitation. Journal of Engineering Physi sand Thermophysi s, 76(4):813�818, 2003.[44℄ S. Launay, V. Sartre, and J. Bonjour. Analyti al model for hara terization of loop heat pipes. Journal of Thermophysi sand Heat Transfer, 22(4):623�631, 2008.[45℄ Yury F. Maydanik, Yury G. Fershtater, and Nikolay N.Solodovnik. Loop heat pipes: Design, investigation, prospe tsof use in aerospa e te hni s. Te hni al Report 941185, SAEInternational, Warrendale, PA, 1994.[46℄ Y. Cao and A. Faghri. Analyti al solutions of �ow and heattransfer in a porous stru ture with partial heating and evapo-ration on the upper surfa e. International Journal of Heat andMass Transfer, 37(10):1525�1533, 1994.14

[47℄ M. Furukawa. Model-based method of theoreti al design anal-ysis of a loop heat pipe. Journal of Thermophysi s and HeatTransfer, 20(1):111�121, 2006.[48℄ Benjamin Siedel, Valérie Sartre, and Frédéri Lefevre. Steady-state analyti al model of a loop heat pipe. In 17th InternationalHeat Pipe Conferen e, Kanpur, India, 2013.[49℄ Joon Hong Boo and Eui Guk Jung. A theoreti al modeling ofa loop heat pipe with a �at evaporator employing the thin-�lmtheory. In 15th International Heat Pipe Conferen e, Clemson,South Carolina (USA), 2010.[50℄ A.A.M. Delil, V. Baturkin, G. Gorbenko, P. Gakal, andV. Ruzaykin. Modelling of a miniature loop heat pipe with a�at evaporator. In Pro eedings of the 32nd International Con-feren e on Environmental Systems, 2002.[51℄ Mohammed Hamdan. Loop heat pipe (LHP) modeling anddevelopment by utilizing oherent porous sili on (CPS) wi ks.PhD thesis, University of Cin innati, 2003.[52℄ Triem T. Hoang and Jentung Ku. Heat and mass transfer inloop heat pipes. In ASME Summer Heat Transfer Conferen e,pages 485�493, Las Vegas, Nevada (USA), 2003.[53℄ Triem T. Hoang, Tamara O'Connell, and Jentung Ku. Math-emati al modeling of loop heat pipes with multiple apillarypumps and multiple ondensers, Part I - steady state simu-lations. In 2nd International Energy Conversion EngineeringConferen e, AIAA Paper 2004-5577, Providen e, Rhode Island(USA), 2004.[54℄ Nima Atabaki. Experimental and omputational studies of loopheat pipes. PhD thesis, M Gill University, Montréal, Québe ,Canada, 2006.[55℄ Nirmalakanth Jesuthasan. Modeling of Thermo�uid Phenom-ena in Segmented Network Simulations of Loop Heat Pipes.PhD thesis, M Gill University, Montréal, Québe , Canada,2011.[56℄ Fang-Chou Lin, Bing-Han Liu, Chi-Ting Huang, and Yau-MingChen. Evaporative heat transfer model of a loop heat pipe withbidisperse wi k stru ture. International Journal of Heat andMass Transfer, 54(21�22):4621�4629, 2011.[57℄ Masakazu Kuroi and Hosei Nagano. The in�uen e of grooveshape on loop heat pipes' performan e. In 16th InternationalHeat Pipe Conferen e, Lyon, Fran e, 2012.[58℄ Lizhan Bai, Guiping Lin, Zuodong Mu, and Dongsheng Wen.Theoreti al analysis of steady-state performan e of a loop heatpipe with a novel evaporator. Applied Thermal Engineering, 64(1�2):233�241, 2014.[59℄ Tao Fang, Tingzhen Ming, C. P. Tso, Xiaoming Huang, andWei Liu. Analysis of non-uniform heat loads on evaporatorswith loop heat pipes. International Journal of Heat and MassTransfer, 75:313�326, 2014.[60℄ Masataka Mitomi and Hosei Nagano. Long-distan e loop heatpipe for e�e tive utilization of energy. International Journal ofHeat and Mass Transfer, 77:777�784, 2014.

15

Authors Year CompleteLHP Partialmodelling

3Danalysis 2Danalysis 1Danalysis Nodalnetwork Analyti alsolution

Cylindri algeometry

Flat(reservoirinthethi kness)

Flat(reservoiradja enttotheev

aporator)Axial/longitudin

alparasiti heat�ux

Radial/transversalparasiti heat

lossesConta tresistan

ebody-wi kCon entratedhe

at�uxConve tioninth

ewi kDryingofthewi

kSe ondary/bipor

ouswi kBayonet Heattransferres

ervoir- asingHard-�lling Finiteevaporatio

n oe� ientIn�niteevaporat

ion oe� ient(settemperature)

Pressuredropinthevapourgroov

esHeattransferin

thevapourgrooves

Longitudinal ondu tioninthetu

besSingle-phasepre

ssuredropsTwo-phasepress

uredropsFlowregime Condensationre

gimeHomogeneousm

odelSeparatedphase

smodelCondensationin

thetransportlines

Gravitye�e ts Non- ondensablegases

Evaporatorheatlosses

Transportlinesheatlosses

Multipleevaporator/ ondenser

Kaya et al. [8℄ 1999 x - - - - x - x - - - x - - - - - - - - - - x - - x - x - - - x x - x x -Figus et al. [31℄ 1999 - x - x - - - x - - - - - - - x - - - - - x - - - - - - - - - - - - - - -Cullimore and Baumann [25℄ 2000 x - - - - x - - - - - x - - x - - - - - - - - - - x x x x x x x x x - x -Delil et al. [50℄ 2002 x - - - - x - - - x x x x - - - - - x - - - - - - x x x - x - - x - x x -Kim and Golliher [26℄ 2002 x - - - - x - - x - - x - - - - - - - - - - - - - x x x x - x - x - x - -Hamdan [51℄ 2003 - x - - x - - - x - - x - - x - - - - - - x - - - - - - - - - - - - - - -Chuang [11℄ 2003 x - - - x - - x - - x x - - x - - - - - - x - x - x x x x - x - x - x x -Hoang and Ku [52℄ 2003 x - - - - x x - - - - x - - - - - x x - x - x - - x x x x - - - - - x x -Hoang et al. [53℄ 2004 x - - - - x - x - - - x - - x - - - - - - - - - - x x x x - x - - - x - xGhajar et al. [28℄ 2005 x - - x - x - - - x x - - - - - - - - - - - x - - x x - - - - - - - - - -Kaya and Goldak [34℄ 2006 - x - x - - - x - - - - - - x x - - - - x - - x - - - - - - - - - - - - -Furukawa [47℄ 2006 x - - - - - x x - - - x - - x - x x - - - x x - - x x - - - x - - - x - -Coquard [32℄ 2006 - x - x - - - x - - - - - - x x - - - - - x - x - - - - - - - - - - - - -Atabaki [54℄ 2006 x - - - - x - x x - - x - - - - - - - - - x x x - x x x x - x x x - x x -Adoni et al. [12℄ 2007 x - - - - x - x x - x x - - x - - - - x - - - - - x x x x - x - - - x x -Chernysheva et al. [24℄ 2007 x - - - - - x - - - - - - - - - - - - x x - x - - x - - - - - - x - - - -Launay et al. [44℄ 2008 x - - - - - x x x - x x - - x - - - - - - - - - - x - x x - - - x - x x -Chernysheva and Maydanik [36℄ 2009 - x - x - - - - - - - - x - - x - - - - - x x - - - - - - - - - - - - - -Adoni et al. [16℄ 2009 x - - - - x - x - - x x - - x - - x - - - - - - - x x x x - x - - - x x - Table1:SummaryoftheLHPst

eady-statemodelsbetween1999

and200916

Authors Year CompleteLHP Partialmodelling

3Danalysis 2Danalysis 1Danalysis Nodalnetwork Analyti alsolution

Cylindri algeometry

Flat(reservoirinthethi kness)

Flat(reservoiradja enttotheev

aporator)Axial/longitudin

alparasiti heat�ux

Radial/transversalparasiti heat

lossesConta tresistan

ebody-wi kCon entratedhe

at�uxConve tioninth

ewi kDryingofthewi

kSe ondary/bipor

ouswi kBayonet Heattransferres

ervoir- asingHard-�lling Finiteevaporatio

n oe� ientIn�niteevaporat

ion oe� ient(settemperature)

Pressuredropinthevapourgroov

esHeattransferin

thevapourgrooves

Longitudinal ondu tioninthetu

besSingle-phasepre

ssuredropsTwo-phasepress

uredropsFlowregime Condensationre

gimeHomogeneousm

odelSeparatedphase

smodelCondensationin

thetransportlines

Gravitye�e ts Non- ondensablegases

Evaporatorheatlosses

Transportlinesheatlosses

Multipleevaporator/ ondenser

Bai et al. [18℄ 2009 x - - - - x - x - - x x - - x - x - x - - x x - - x x x x - x x - - x x -Singh et al. [20℄ 2009 x - - - - x - - x - - x - - - - - - - - - - - - - x - x - - - - x - - - -Hamdan and Elnajjar [30℄ 2009 x - - - - x - - x - - x - - x - - - - - - - - - - x - - - - - - - - - - -Adoni et al. [17℄ 2010 x - - - - x - x x - x x - - x - - x - x - - - - - x x x x - x - - - x x -Rivière et al. [21℄ 2010 x - - - x x - x - - x x - - x - - - - - - - x x x x x x x - x x - - x x -Boo and Jung [49℄ 2010 x - - - - - x - x - - x - - x - - - - x x - - x - - - - - - - - - - x x -Li and Peterson [37℄ 2011 - x x - - - - - x - - - - - x - - - - - - x x x - - - - - - - - - - - - -Jesuthasan [55℄ 2011 x - - - x x - x x - - x - - - - - - - - - - x x - x x x x - x x x - x x -Lin et al. [56℄ 2011 x - - - - x - x - - - x x - - x x - - - - - - - - x x - - - x - - - x - -Zhang et al. [39℄ 2012 - x x - - - - - - x - x - - x - - - - - - x x x - x - - - - - - - - - - -Kuroi and Nagano [57℄ 2012 x - - - x x - x - - x x - - - - - - - - - - - x - x x x - - x - - - x - -Chernysheva and Maydanik [40℄ 2012 - x x - - - - - - x x x x - x x - - x - x - x x - - - - - - - - - - x - -Chernysheva and Maydanik [41℄ 2012 - x x - - - - - - x x x x x x x - - x - x - x x - - - - - - - - - - x - -Hodot et al. [23℄ 2013 x - x - x - - x - - x x - - x - - x x - x - - x x x x x x - x x x - x x -Siedel et al. [22℄ 2013 x - - x x - - - x - x x x - x - - - x - x - x x x x x x x - x x x x x x -Siedel et al. [48℄ 2013 x - x x - x x x x - x x - - x - - - x - x - - - - x - x - - - - x x x x -Bai et al. [58℄ 2014 x - - - - x - x - - x x - - - - - - x x - x - - - x x x x - x x x - x x -Fang et al. [59℄ 2014 - x - x - - - - - x x x - x x x - - - - x - - x - - - - - - - - - - - - -Mitomi and Nagano [60℄ 2014 x - - - x x - x - - x x x - - - - - - - x - x x - x x x - - x x - - x x - Table2:SummaryoftheLHPst

eady-statemodelsbetween2009

and201417