lo

, Guiping Lin a, Jiang He b, Dongsheng Wen c

a Laboratory of Fundamental Science on Ergonomics anBeijing 100191, PR Chinab Beijing Key Laboratory of Space Thermal Control Techc School of Chemical and Process Engineering, Universit

vity-ased andxhibitsHP perak to th

Received 12 November 2014Accepted 13 March 2015

terrestrial applications. This work addresses our insufcient understanding of LHP operation undergravity-assisted attitude, i.e. the condenser is located higher than the evaporator. A steady-state math-

ecially for terrestrial

. All rights reserved.

tion of a working uid to transfer heat, and the capillary forcesdeveloped in ne porous wicks to circulate the working uid [1,2].Their high pumping capability and superior heat transport perfor-mance have been traditionally utilized to address the thermal-

were successfullytly, its application

submarines [12e15]. Their long distance heat transport capabilityand exibility in design could offer many advantages comparedwith traditional heat pipes and other heat transfer devices.

So far, quite a few studies on themathematical modeling of LHPshave been conducted, which revealed someworking principles andoperating characteristics of LHPs [16e23], as briey reviewedbelow. Kaya et al. [16] established a one-dimensional steady-state* Corresponding author. Tel./fax: 86 10 8233 8600.

Contents lists available at ScienceDirect

Applied Therma

ev

Applied Thermal Engineering 83 (2015) 88e97E-mail address: bailizhan@buaa.edu.cn (L. Bai).Loop heat pipes (LHPs) are effective and efcient two-phaseheat transfer devices that utilize the evaporation and condensa-

has been extended to terrestrial surroundings such as in electronicscooling [8e11] and thermal-management systems for aircraft andunderstanding of the complicated operating principle and characteristics of LHPs espapplications.

2015 Elsevier Ltd

1. Introduction management problems of spacecraft, and theyapplied in many space tasks [3e7]. More recenOperating characteristicsExperiment

mass owrate in the loop shows a unique V-shape with the increase of applied heat load; ii) the steady-state operating temperature is much lower under the gravity driven mode, and is in similar values undercapillarity-gravity co-driven mode and iii) the thermal conductance of the LHP increases with increasingpositive elevation especially in the variable conductance zone. Such results contribute greatly to theAvailable online 21 March 2015

Keywords:Loop heat pipeMathematical modelDriving mode

ematical model of a LHP under gravity-assisted operation was established based on two driving modes:gravity driven mode and capillarity-gravity co-driven mode, determined by a dened transition heatload. The model was validated by the experimental results, and was employed to predict the operatingcharacteristics of a LHP under the gravity-assisted attitude. Comparing to LHPs operating under hori-zontal or antigravity attitudes, some distinctive features have been identied, which include: i) the totalh i g h l i g h t s

Steady-state model of a LHP under gra Two driving modes have been identi Operating temperature curve of LHP e Positive elevation has great effect on L Enhanced cooling and reduced heat le

a r t i c l e i n f o

Article history:http://dx.doi.org/10.1016/j.applthermaleng.2015.03.011359-4311/ 2015 Elsevier Ltd. All rights reserved.d Environmental Control, School of Aeronautic Science and Engineering, Beihang University,

nology, Beijing Institute of Spacecraft System Engineering, Beijing 100094, PR Chinay of Leeds, Leeds LS2 9JT, UK

sisted operation has been established.theoretically analyzed.unique trend under gravity-driven mode.formance under gravity-driven mode.e CC lead to better LHP performance.

a b s t r a c t

Loop heat pipes (LHPs) are efcient two-phase heat transfer devices that have found many space andLizhan Bai a, *, Jinghui Guo agravity-assisted operationResearch paper

Steady-state modeling and analysis of a

journal homepage: www.els4op heat pipe under

l Engineering

ier .com/locate/apthermeng

al Enmathematical model of a LHP, which could reect the variableconductance characteristics of the LHP, but the oversimplicationin calculating the radial conductance of the wick and two-phasepressure drop in the condenser brought large difference betweenthe modeling and experimental results. Improved treatment on thetwo-phase pressure drop in the condenser was conducted byHoang et al. [17], where ve different two-phase pressure dropcorrelations were assessed and better predictions were achieved.The modeling of the radial conductance of the evaporator wick wasimproved by Parker [18] through solving a radial one-dimensionalenergy equation, in which the effect of the uid convection wasconsidered. Chuang et al. [19] developed a comprehensive steady-state model, which showed good predictions when the evapo-rator was horizontal with or higher than the condenser, but largedeviation from the experimental results was observed under thegravity-assisted attitude. Further improvement was attempted byVlassov et al. [20] where the liquid/vapor interface in the condenserand the void fraction in the compensation chamber (CC) could bedetermined. Bai et al. [21] developed a steady-state mathematicalmodel of a LHP by considering the evaporator wick as either single-layer or two-layer composite structures, and the condenser havingthe annular ow pattern. The effects of surface tension of liquid andthe interaction between the liquid and vapor phases in thecondenser including both frictional and momentum-transfer shearstresses were considered. The model revealed the observed ther-mal conductance reduction phenomenawhen LHPs operated in theconstant conductance zone, which cannot be reected by tradi-tional models. Fang et al. [22] conducted a numerical analysis basedon a two-dimensional dynamic mesh model to investigate the in-uence of non-uniform heat load on the performance of a at-plateevaporator LHP. The variations of evaporation heat transfer coef-cient, outow working uid temperature, vapor and liquid inter-face position and surface temperature at different heat loads wereanalyzed. A mathematical model of the startup process of a LHPwas established by Bai et al. [23] based on the node networkmethod, and a parametric analysis including the effects of startupheat load, thermal capacity of the evaporator and CC, heat sinktemperature and ambient temperature on the startup characteris-tics of the LHP were conducted.

To the best of our knowledge, the mathematical models pre-sented above are only applicable to the situation when the LHP isunder horizontal or antigravity attitudes. When the LHP is oper-ating under the gravity-assisted attitude, i.e. the condenser islocated higher than the evaporator, large deviation will exist be-tween the modeling and experimental results. Chuang et al. [24]conducted an experimental and analytical study of a loop heatpipe at a positive elevation using neutron radiography. The changeof the liquid/vapor distribution and ow pattern in the vaportransport line under the gravity-assisted attitude was captured bythe employment of the non-destructive visualization technique.Experimental results showed that when the LHP was operatingunder the gravity-assisted attitude, the operating temperatureexhibited unique variation trend, and the authors proposed agravity-assisted operating theory, which categorized the steady-state operation into capillary-controlled and gravity-controlledmodes to better explain the observed experimental phenomena.

However, theoretical investigation on the LHP operation underthe gravity-assisted attitude is quite limited and obviously inade-quate so far, i.e. in the experimental and analytical study of Ref [24],the positive elevation range is quite small (0e127 mm), and theheat sink temperature is set as a xed value of 5 C, which cannotreect the unique characteristics at a relatively large positiveelevation and lower heat sink temperature; in addition, how thetransition heat load separating the LHP operation into the capillary-

L. Bai et al. / Applied Thermcontrolled and gravity-controlled modes changes with operatingparameters such as the positive elevation and heat sink tempera-ture is still not well understood.With the rapid development of LHPapplication in terrestrial surroundings, it is of great interest toestablish an accurate mathematical model of LHPs operating underthe gravity-assisted attitude to better understand its operatingprinciple and characteristics and guide the engineering design,which forms the objective of this study.

2. Mathematical modeling

2.1. Two driving modes

When the LHP is operating under horizontal or antigravity atti-tudes, the capillary force generated by the evaporator wick is thedriving source for the circulation of the working uid in the loop.However, when the LHP is operating under the gravity-assisted atti-tude, the situation becomes much different and very complicated.

As reviewed above, two driving modes have been identiedunder the gravity-assisted attitude [24,25]: gravity-driven modeand capillarity-gravity co-driven mode, depending on the appliedheat load. At a small heat load, the LHP tends to operate in thegravity-driven mode where the gravity is the only driving sourcefor the circulation of the working uid. Under such a condition, theworking uid in the vapor line is in the two-phase state due to theexistence of additional liquid mass ow, and no clear liquid/vaporinterface exists at the outer surface of the evaporator wick, asshown in Fig. 1(a). At a relatively large heat load, the LHP operatesin the capillarity-gravity co-driven mode, where the capillary forceand gravity are both driving sources for the circulation of theworking uid. Under this condition, the working uid in the vaporline is pure vapor, and there is a clear liquid/vapor interface at theouter surface of the evaporator wick, as shown in Fig. 1(b).

2.2. Determination of the transition heat load

To realize the steady-state modeling of a LHP under gravity-assisted operation, the rst and most important step is to deter-mine the transition heat load (Qtr), i.e. the heat load responsible forthe transition from the gravity driven mode to capillarity-gravityco-driven model. With the transition heat load applied to theevaporator, the working uid in the vapor line is pure vapor, andthe gravitational pressure head generated in the liquid line justsatises the requirement to drive the circulation of the workinguid in the loop, so the pressure balance equation can be expressedas:

Zll

rlgdH DPvg DPvl DPc DPll DPwi (1)

Because the gravitational pressure head in the liquid line andthe frictional pressure drop in each component of the LHP are bothstrong functions of the operating temperature, however, thesteady-state operating temperature is initially unknown, it isimpossible to directly calculate the transition heat load based onEquation (1). At the same time, it reminds us that the transitionheat load and the steady-state operating temperature should beobtained simultaneously.

In order to obtain the transition heat load and the operatingtemperature simultaneously, a detailed solution owchart is pre-sented in Fig. 2. Below are some introductions to the solutionowchart:

1) Calculation of the heat transfer and pressure drop in eachcomponent of the LHP in Fig. 2 is the same as that in our pre-

gineering 83 (2015) 88e97 89vious mathematical model [21], and is not repeated here.

l EnL. Bai et al. / Applied Therma902) In the solution process, a relatively large heat load is rstlyspecied as the transition heat load, which is larger than theactual one. In general, the specied relatively large heat load canbe simply determined based on the condition that the gravita-tional pressure head in the liquid line where the liquid is at thelowest temperature (the heat sink temperature) is equal to thefrictional pressure drop in the vapor line.

3) By decreasing the transition heat load gradually and calculatingthe temperature distribution along the loop, the total pressuredrop along the loop will be just balanced by the gravitationalpressure head generated in the liquid line. Under this condition,the actual transition heat load and the steady-state operatingtemperature can be obtained simultaneously.

2.3. Gravity driven mode

When the heat load applied to the evaporator is smaller than thetransition heat load, LHP operates in the gravity driven mode. Thegravitational pressure difference between the liquid line and vaporline is the driving source for the circulation of the working uid inthe loop, so the pressure balance equation can be expressed as:

Fig. 1. Working uid distribution inside LHP for different driving modes.Zll

rlgdH Zvl

rgdH DPvg DPvl DPc DPll DPwi (2)

where r rva rl1 a (3)

a xrlxrl 1 xrvS

(4)

Generally, the vapor mass owrate in the vapor line in thegravity driven mode is relatively small, and the ow pattern in thevapor line is mainly bubbly or slug ow. For simplication purpose,homogeneous owmodel is adopted here, and the slip ratio (S) canbe set as 1.0 accordingly. By substituting Equation (4) into Equation(3), the average density of the working uid in the vapor line can beexpressed as:

r rlrvxrl 1 xrv

(5)

In the gravity drivenmode, as theworking uid in the vapor lineis in the two-phase state, the actual mass owrate of the system atthe inlet of the vapor line becomes:

_m _mv _ml (6)Based on the energy conservation, the vapor mass owrate at

the inlet of the vapor line can be expressed as:

_mv Qap Qhl Qhwl

(7)

The vapor mass owrate is proportional to the heat load appliedto the evaporator at steady-state conditions. Because the pressuredrop in each component is directly related to themass owrate, theliquid mass owrate at the inlet of the vapor line naturally adjustsitself to match the pressure balance as shown in Equation (2) undera steady-state condition.

Because the working uid in the vapor line is in the two-phasestate, the heat transfer between the working uid in the vapor lineand the ambient is in the form of latent heat, and the energyequation of the vapor line can be expressed as follows by neglectingthe small thermal conduction through the pipe wall:

_mdhdL

G=LvlaT Ta (8)

As the enthalpy value depends on the selection of the initialstate at which the enthalpy is zero, the enthalpy of the saturatedliquid with respect to the local pressure is set zero here, and theenthalpy value of the working uid at different states then can beexpressed as:

h 8 TsatT TsatT < Tsat

superheated vaportwo phase statesubcooled liquid

(9)

Accordingly, the temperature and vapor quality at differentstates can be calculated as:

T 8