3D heat transfer analysis in a loop heat pipe evaporator wit

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

Ji Li a ,?,G.P.Peterson b

a

Laboratory of Electronics Thermal Management,College of Physics,Graduate University of the Chinese Academy of Sciences,19A Yu-quan-lu Road,Shijingshan District,Beijing 100049,PR China b

The G.W.Woodruff School of Mechanical Engineering,Georgia Institute of Technology,Atlanta,GA 30332,United States

a r t i c l e i n f o Article history:

Received 4May 2010

Received in revised form 30July 2010Accepted 9September 2010

Available online 15October 2010Keywords:

Loop heat pipe

Numerical simulation Three dimensional Heat transfer Mass transfer

a b s t r a c t

A practical quasi three-dimensional numerical model is developed to investigate the heat and mass trans-fer in a square ?at evaporator of a loop heat pipe with a fully saturated wicking structure.The conjugate heat transfer problem is coupled with a detailed mass transfer in the wick structure,and incorporated with the phase change occurring at the liquid–vapor interface.The three-dimensional governing equa-tions for the heat and mass transfer (continuity,Darcy and energy)are developed,with speci?c attention given to the wick region.By comparing the results of the numerical simulations and the experimental tests,the local heat transfer mechanisms are revealed,through the obtained temperature distribution and the further derived evaporation rates along the liquid–vapor interface.The results indicate that the model developed herein can provide an insight in understanding the thermal characteristics of loop heat pipes during steady-state operation,especially at low heat loads.

Ó2010Elsevier Ltd.All rights reserved.

1.Introduction

Two-phase capillary pumped heat transfer devices are becom-ing much more prevalent in the thermal control of electronic de-vices,particularly in the increasingly demanding thermal control problems of high-end electronics.Among these devices,loop heat pipes (LHP)are particularly interesting,due to the advantages,which include:(i)greatly increased capacity over conventional heat pipes at comparable dimensions;(ii)more ef?cient operation at any orientation in the gravitational ?eld;(iii)lower overall ther-mal resistance;(iv)increased ?exibility in packaging;and (v)high heat load capacities over longer distance [1].A loop heat pipe (LHP)is a two-phase heat transfer device that removes heat from a source (e.g.,an electronic chip)through an evaporator and pas-sively moves it to a condenser region or radiator using capillary forces to pump the ?uid,ultimately releasing the heat to the envi-ronment from the condenser by natural or forced convection.Loop heat pipes were ?rst investigated and patented in the USSR in 1979by Kiseev et al.[1,2].The patent for LHPs was ?led in the USA in 1982[1,2].There were many different con?gurations of LHP evap-orators and different types (homogeneous or heterogeneous)of wicks,which were investigated to improve the performance of LHPs and determine the compatibility of the materials.A detailed description of the operating characteristics,working principles and persity of LHP con?gurations can be found in Maidannik [2],Ku [3],and Launay et al.[4],along with a parametric analysis of the operations of LHPs.A performance comparison for miniatur-ized or compact LHPs can be found in Li et al.[5,6].

Historically,theoretical analyses or numerical simulations in-volved in the operation of LHPs has lagged far behind the experi-mental investigations,due to the complexity of the operational modalities and the complexity of the problems involved.On the basis of the pioneering theoretical works on traditional heat pipes (and capillary pumped loops),Ku [3]?rst developed an analytical model to describe the thermal-hydraulic behaviors of the LHP and described the operating characteristics and performance limi-tations as a function of the various physical and operational parameters,including the heat load,sink temperature,ambient temperature,and elevation change.Then,Chernysheva et al.[7]proposed a semi-empirical theoretical method to calculate the LHP operating temperature for both fully and partially saturated compensation chambers and discussed the different features and operational advantages and disadvantages for these two operating modes.Most recently,Bai et al.[8]established a mathematical model considering the in?uence of the compound wick over the performance of LHPs coupled with the annular ?ow model for con-denser.Only a limited number of numerical investigations have been presented for LHP evaporators [9–11].Cao and Faghri [9]pre-sented a numerical analysis for a completely liquid saturated wick in a capillary pumped loop with a ?at-plate type evaporator,in which a two-dimensional liquid ?ow in the wick and a three-dimensional vapor ?ow in the grooves separated by the wick were considered.By adopting a numerical model for heat pipes with an inverted meniscus developed from Demidov and Ystsenko [12]and later Figus et al.[13],as depicted in Fig.1,Kaya and Goldak [10]

0017-9310/$-see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/j.ijheatmasstransfer.2010.09.014

Corresponding author.Tel.:+86135********.

E-mail address:jili@ (J.Li).

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

conducted a two-dimensional numerical analysis of heat and mass transfer in the wick of a loop heat pipe with a cylindrical evapora-tor based on the results of previous works[9,12,13],e.g.,the valid-ity of the Darcy law for a wick with a varying pore-size distribution,and the accuracy of a two-dimensional model for heat and mass transfer in the porous structure.They reported that the overall temperature distribution is not strongly affected by the choice of the correlation used to predict the effective thermal con-ductivity of the wick.In this model,only the heat and mass transfer in the wick and the evaporator cover were given consideration.An empirical correlation was used to de?ne the boundaries of the sur-formation of the meniscus are desirable.Inspired by the concept of mini-/micro-channel heat transfer and by the fundamental?nd-ings reported by Li et al.[14],Li and his co-workers?rst proposed a unique square,?at LHP evaporator with a wicked?n directly sin-tered onto the substrate(or casing)of the LHP evaporator[5,6], see Fig.2.With this novel design of the porous structure,an extre-mely low thermal resistance in the LHP evaporator was obtained [5,6].The mechanism behind this phenomenon was also carefully examined[6]and it is believed that Demidov and Ystsenko’s argu-ment is correct.

Based upon the analysis above,through a thorough literature

Nomenclature

c p speci?c heat,J/kg K

h fg latent heat of evaporation,J/kg

K p permeability,m2

k thermal conductivity,W/m K

k e effective thermal conductivity,W/m K

_m mass?ow rate,kg/s

n number of vapor removal channels

P pressure,Pa

Q load total heat load

R g gas constant,J/kg K

T temperature,°C

Greek symbols

a heat transfer coef?cient,W/m2K

C border of wick and vapor removal channel m kinetic viscosity,m2/s

q density,kg/m3

e porosity Subscripts

a ambient

c condenser

cap capillarity

cc compensation chamber

cu copper

e evaporator

g gravity

l liquid

ll liquid line

n normal direction

q heat?ux

s substrate;solid phase material in wick

v vapor

vl vapor line

vc vapor removal channel

w wick

Fig.2.Schematic structure of a unique square?at LHP evaporator with wicked?n directly sintered on the substrate[5,6].

J.Li,G.P.Peterson/International Journal of Heat and Mass Transfer54(2011)564–574565

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

et al.[7]into the determination of the saturation pressure of the working?uid in the compensation chamber as a predetermined

(1)the process is steady-state,

(2)the capillary structure is homogeneous and isotropic,

(3)the radiative and gravitational effects are negligible,and

(4)the?uid is Newtonian and has constant properties at each

phase.

In addition,there are other assumptions in the present model involving the boundary conditions,the properties of the porous structure and the liquid–vapor interface,which are listed below:

(1)the wick structure is perfectly saturated,

(2)the liquid-vapor interface has zero thickness,

(3)the sharp discontinuities of the properties are maintained

across the interface,

(4)the temperature at the liquid-vapor interface(here the wick

interface)is the saturation temperature corresponding to the local static pressure,

(5)with the exception of the bottom of the evaporator substrate

and the top surface of the liquid in the compensation cham-ber,all other thermal boundary conditions are chosen as adi-abatic condition.

The above assumptions are mostly identical to those used by Cao and Faghri[9],and Kaya and Goldak[10].The problem studied here is a conjugate heat transfer problem coupled with mass trans-fer in the compensation chamber(liquid phase),the vapor removal channel(vapor phase),and in the wick structure(liquid phase), and also coupled with phase change taking place at the liquid–vapor interface(evaporation at the wick border C,referring to Fig.3).The governing equations for the heat and mass transfer (continuity,Darcy and energy)can be summarized as follows: Governing equations:

Liquid?ow and heat transfer in the compensation chamber

u¼0;v¼À

_m

l

q

l

A cc

;w¼0;ð1Þ

v@T

@y

¼

k l

q

l

Ác p;l

@2T

@x2

þ

@2T

@y2

þ

@2T

@z2

!

ð2Þ

here A cc¼L pitchÂL e,and_m l is the liquid mass?ow rate per pitch.

Vapor?ow and heat transfer in the vapor removal channel

@uðxÞ

@x

¼À

1

W cÁH c

Z

C

1

q vÁh fg k w

@T

@n

CÀ

Àk v

@T

@n

Cþ

!

d C;

v¼0;w¼0;ð3Þu

@T

¼

k v

q v p;v

@2T

þ

@2T

þ

@2T

!

:ð4ÞLiquid?ow and heat transfer in the wick

@u

@x

þ

@v

@y

þ

@w

@z

¼0;ð5Þu¼À

K p

l

l

@P w

@x

;v¼ÀK p l

l

@P w

@y

;w¼À

K p

l

l

@P w

@z

;ð6Þu

@T

@x

þv@T@yþw@T@z¼k w

q

l

Ác p;l

@2T

@x2

þ

@2T

@y2

þ

@2T

@z2

!

:ð7Þ

The adequacy of the Darcy law to describe the?ow inside the wick has been discussed extensively by Kaya and Goldak[10], especially for low heat?ux rates.With the boundary conditions (hydrodynamic and thermal)shown in Table1,the problem is mathematically closed and the pressure,velocity and the

566J.Li,G.P.Peterson/International Journal of Heat and Mass Transfer54(2011)564–574

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

temperature can be solved numerically.A temperature boundary condition at the liquid-vapor interface is adopted here due to the assumption of a fully saturated wick.Kaya and Goldak [10]pro-posed the following interfacial conditions to identify the vapor–liquid interface in the wick,

The mass balance at C ,

ðu n Þv Áq v ¼ðu n Þl Áq l :

ð22Þ

And the energy balance at C ,

Àk w @T C

ÀÀk v @T C

¼ðu n Þl Áq l Áh fg ¼ðu n Þv Áq v Áh fg :ð23

Þ

Fig.4.Geometric parameters of the numerical domain.

Table 1

Boundary conditions.

At x =0,

for wick region and compensation chamber region

@P @x ¼0;@u @x ¼0;@v @x

¼0;@w @x ¼0;@T

@x ¼0ð8Þ

for vapor region

P v ;e ðx ¼0Þ¼exp

25:511À1:065Áh fg =R g

e ;v ð9Þ

u ¼0;

v ¼0;

w ¼0;

@T @x

¼0ð10Þ

At x =L ,

for wick region and compensation chamber region

@P @x ¼0;@u @x ¼0;@v @x

¼0;@w @x ¼0;@T @x ¼0ð11Þ

for vapor region

P v ;e ðx ¼L Þ¼exp 25:511À1:065Áh fg =R g

T e ;v þ273:15

ÀD P v c

ð12Þ

u ðx ¼L e Þ¼À1c c Z x ¼L e x ¼0Z

C 1q v fg k w @T C À Àk v

@T @n C þ

!d C dx ;v ¼0;

w ¼0;@T ¼0ð13Þ

At y =0(solid boundary conditions)

Àk cu

@T

@y

¼q ð14Þ

At y =H (liquid phase),

T ¼T l ;cc ¼T s ðP l ;cc Þ

ð15Þ

At z =0

@P @z ¼0;@u @z ¼0;@v @z

¼0;w ¼0;@T @y ¼0ð16Þ

At z =L pitch /2

@P @z ¼0;@u @z ¼0;@v @z

¼0;w ¼0;@T

@y ¼0ð17Þ

At the inner wick boundaries,At y =d ,

@P

@y

¼0;u ¼0;v ¼0;

w ¼0ð18Þ

At y =H w ,

P ¼P l ;cc ;u ¼0;

v _¼À

_m

l q l A w

;w ¼0ð19Þ

here,A w %L pitch ÂL e Âe

At C ,

u n ¼À

1

q l h fg

k w @T @n C ÀÀk v @T @n C þ

!

;@P @n C

¼À

l l

K p

u n ;

ð20Þ

T n ¼T s ½P v ðx Þ ;P v ðx Þ¼P v ;e ÀD P v c

x

L e

ð21Þ

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

The above considerations have been

model along with the boundary and (20)for the completely saturated is partially saturated,not as described by boundary conditions Eqs.(13)and (20)but the position of the interface will no boundary,C .This situation will need to in future studies along with the vapor ?ow should be added to the model,in a ?ow controlled by the Darcy law.

Through careful examination of the Eqs.(1)–(21),one may be curious in how saturation pressure P v ,e as it appears in Eq.sure in the compensation chamber P l ,cc in conductivity of the liquid saturated wick critical parameters that govern the ?nal P v ,e is the vapor pressure in the vapor be approximated from the vapor the Clausius–Clapeyron equation.The posed by Chernysheva et al.[7]can be used temperature in the vapor removal channel der a given heat load and a detailed presented in Appendix A .

The permeability of the porous (20)can be calculated directly from the equation [15]for the sintered spherical K p ¼

d 2

powder Áe 3150ð1Àe Þ

:Determination of the effective thermal conductivity of the li-quid saturated wick utilized several well-accepted models for sin-tered spherical metal powder:Gorring and Churchill’s model

k w ¼

k l ½ð2k l þk s ÞÀ2ð1Àe Þðk l Àk s Þ

l s e l s ;

ð25Þ

where k s is the thermal conductivity of the wick material (i.e.,copper).

Maxwell’s model

k w ¼k s 2þk l =k s À2e ð1Àk l =k s Þ

2þk l =k s þe ð1Àk l =k s Þ

:

ð26Þ

Chaudary and Bhandari’s model

k w ¼ðk max Þn Áðk min Þ1Àn ;

ð27Þ

where

k max ¼e Ák l þð1Àe ÞÁk s ðparallel case Þ;k min ¼ðk l Ák s Þ=½e Ák s þð1Àe ÞÁk l ðseries case Þ:

From other literatures [16,17],it was found that Gorring and Churchill’s model underestimated the value of k w ,while Maxwell’s model over-predicted the value of k w ,by several times of the value as measured by Singh [16].Chaudary and Bhandari’s model has been demonstrated to be appropriate for the present type of wick structure when n =0.42[8,17].

All the necessary property information associated with the wick is shown in Table 2,as determined by Eqs.(24)and (27)with the aid of experimental measurements for the copper powder wick.Based on the data reported in the literature [18],the SEM measure-ments as shown in Fig.5,and the weight analysis method,for the present wick structure,the porosity was determined to be 50%for the sintered powder with mesh numbers between 100and 140(it should be noted here that for such a wick,the measured porosity as determined from SEM measurements was between 50%and 60%,

depending on the various powder densities or different sintering forces used during the wick preparation process.However,using a comparative weight analysis method,which compares the mea-sured weight of the fabricated wick and the calculated weight of the wick from its material density and volume,the measured data is close to the data reported in Ref.[18],i.e.,approximately 40%).3.Numerical procedure

The governing partial differential equations with the boundary conditions were discretized using a control volume method with grid points placed at the center of each cell.The upwind scheme was used for the convection term in Eq.(2),Eqs.(4)and (7),and were solved by marching downstream.In order to expedite the convergence of the calculations,a line-by-line iteration and a Tri-diagonal Matrix Algorithm (TDMA)along with a Thomas algorithm solver,and successive under-relaxation iterative methods were used to obtain the three-dimensional temperature distribution,T (x ,y ,z ).The overall numerical procedure is illustrated in Fig.6.The convergence criteria were established as:

X X X

j P w ;new ði ;j ;k ÞÀP w ;old ði ;j ;k Þj 610À3

ð28Þ

for pressure the distribution in the wick and

X X X

j T new ði ;j ;k ÞÀT old ði ;j ;k Þj 610À2

ð29Þ

for the temperature distribution in the domain.

The total grid number is 84,000(i (x )Âj (y )Âk (z )is 50Â56Â30)for the domain as shown in Fig.4.Eq.(29)yields an approximate criterion for the mean square root error (MSRE)

of j T ði ;j ;k ÞÀT 0ði ;j ;k Þj %PPP ???????????????????????????

ðT ði ;j ;k ÞÀT 0ði ;j ;k ÞÞ2

p 61:2Â10À7.This type of a ?ne grid mesh for the x ,y and z directions was chosen in order to properly resolve the boundary conditions,and to better de?ne the conjugate heat transfer at the surface of the channel,thereby improving the temperature resolution.At the border of the different regions,the interface thermal conductivity is set as

568J.Li,G.P.

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

ature–volume diagram.

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

Fig.8.Temperature distribution in the LHP evaporator for q=15W/cm2.

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

also indicates that the majority of the heat is absorbed and deliv-ered through the evaporation of the working ?uid,and only a very

corresponding to Fig.8,which was obtained through numerically solving the Laplace equation r 2P ¼0,which comes from combin-J.Li,G.P.Peterson /International Journal of Heat and Mass Transfer 54(2011)564–574571

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

averaged magnitude of the pressure drop(?ow

wick are2.75Pa,9.1Pa,and18.9Pa for q=5W/cm2

30W/cm2,respectively from the computations.

Correspondingly,the velocity vector?eld in the

trated in Fig.10in the y–z plane at x=L e/2for q=15

was obtained from Eq.(6)with the boundary

in Eqs.(18)–(21)after the pressure distribution in

obtained.

In order to help in understanding the heat transfer

along the liquid–vapor interface inside the

tion of the heat?ux distribution is given in Fig.11.The

?ux occurs at the wick border where it contacts

the substrate and the heat?ux decreases along the

wicked?n quickly.This numerical observation

and Ystsenko’s argument as being pointed out in introduction.

In order to verify the accuracy of the numerical

ability to accurately predict the performance of LHPs,a

of the simulation results and the experimental data was per-

formed.A detailed description of the test facility and the various parameters utilized during the LHP operation have been reported previously[5].

It should be noted that for a given heat load,with the aid of the semi-empirical theoretical model given by Chernysheva et al.[7] (as partially summarized in Appendix A),the saturation status at the top surface of the liquid in the compensation chamber and at the vapor removal channel was predetermined quantitatively at the start of the numerical simulation.Fig.12shows the theoret-ical calculations of saturation status of the working?uid inside the loop and the comparison with the experimental measurements at different places on the external surface of the tested loop heat pipe for q=15W/cm2with an illustration of a schematic of the loop heat pipe in test(here?ve microscale T-type Omega thermocou-ples were?rmly attached onto the loop heat pipe by?lm adhesive as shown,to measure:the temperature at the bottom surface of the evaporator(102);the temperature at the exit of the evaporator (103);the temperature at the entrance of the condenser(104);the temperature at the exit of the condenser(105);the temperature at the compensation chamber.However,at the present time,the in?uence from the above two matters are very dif?cult to evaluate.

Fig.13illustrates a comparison between the numerical results for the bottom surface temperature of the LHP evaporator and the experimental results under the same operation parameters (referring to[5,6]).In Fig.13,the dotted line shows the experimental results of the average temperature at the bottom surface of the LHP evaporator for Q load=30W,100W,and200W,respectively,and the starred line shows the numerical results at the bottom surface of the LHP evaporator for Q load=31.25W(q=5W/cm2),93.75W (q=15W/cm2),and187.5W(q=30W/cm2),respectively.

If we take into account the numerical solution error,the exper-imental uncertainties,and the limitations of the present model,the present numerical approach correlates quite well,particularly in the region surrounding the point where q=15W/cm2.It can be in-ferred from these results that in this region,the wick satis?es the assumption of a perfectly saturated wick,which was also con-?rmed by a previous investigation[6]for Q load=100–150W after the LHP reached steady-state operation.Besides the reasons as dis-cussed above,the disparity between the numerical results and the experimental measurements at heat loads other than q=15W/cm2 can be pided into two different situations:First,if the heat?ux is smaller than the value required to force all of the liquid inside the vapor removal channels out,large oscillations in the LHP tempera-ture during operation will occur[5,6].The blockage of the vapor removal channels can signi?cantly degrade the performance of the loop heat pipe;second,if the heat?ux is higher than the value required to maintain the super?cial liquid–vapor interface at the wick-channel border,a liquid–vapor interface will be formed inside the wick and the evaporation interface area will be enlarged. This will result in an enhanced heat transfer mechanism in the evaporator.A new computational methodology is currently being developed that will be able to identify the evolution process of the liquid–vapor interface in the wick automatically for different heat loads.

5.Conclusions

In this work,a practical quasi-3D numerical model for?uid?ow and heat transfer in a?at square LHP evaporator was developed and the local heat transfer mechanism inside the LHP evaporator was determined by analyzing the simulation results.The results of the numerical model compare quite well with the available experimental results for the fully saturated operational mode in the wick at low heat loads.Based on the present model,further

Fig.12.Theoretical calculations of?uid temperature inside the loop and their comparison with the experimental measurements at different places on the external surface of the tested loop heat pipe under q=15W/cm2.

Case temperature comparison between the numerical simulations experimental measurements for different heat loads.

572J.Li,G.P.Peterson/

3D heat transfer analysis in a loop heat pipe evaporator with a fully saturated wick

geometric optimization in term of the performance of loop heat pipes is possible for a speci?c heat load.

Acknowledgements

This work was partially supported by the President Fund of Graduate University of the Chinese Academy of Sciences (085101AM03)and partially supported by National Science and Technology Ministry(2009GB104001).

Appendix A.Determination of saturated pressures and temperatures inside the loop

Referring to Fig.7,given a certain heat load,the mass?ow rate of liquid in a loop heat pipe is,

_m¼Q load

h fg

:ðA:1Þ

It is well known that the pressure losses in the loop should be balanced by the capillary force in order to maintain continuous operation of the loop heat pipe,

D P cap¼D P vþD P lþD P wÇD P g:ðA:2Þ

The pressure loss for the vapor phase in the loop can be calcu-lated from

D P v¼

X

i

D P i¼D P v cþD P v lþD P v;condenser:ðA:3Þ

The terms in the right side of the equation above takes into ac-count the pressure drop in the vapor removal channels(rectangu-lar shape)in the evaporator D P vc,in the vapor line(circular pipe) D P vl,and in the vapor section of the condenser(circular pipe) D P v,condenser respectively and can be evaluated from the Fanning friction equation by substituting different friction factors in the equation for the different geometries respectively,

In the vapor removal channel,

D P v c¼2 u v

c

ÁcÁl vÁl v c

d v

c

;ðA:4Þ

where u v c¼_m

v c q v ,c¼4:7þ19:641þa2

ð1þÞ

,and a¼H.

In the vapor line,

D P v l¼128Ál v lÁt vÁ_m

Ád v

l

:ðA:5Þ

And in the vapor section of the condenser,

D P v;condenser¼128Ál v;cÁt vÁ_m

pÁd

c

:ðA:6Þ

For the pressure loss of the liquid phase in the loop,

D P l¼P l;cÀP l;cc¼128Ál llÁt lÁ_m

pÁd

ll

:ðA:7Þ

For the pressure loss in the wick during liquid?ow,

D P w¼P l;ccÀP l;w¼l

l

Á u l;wÁd w

p

Darcy Equation;ðA:8Þ

where u l;w¼_m

q

l w

.

And?nally,from the gravity,

D P g¼ðq lÀq vÞgh:ðA:9Þ

Thus,all of the terms in Eq.(A.2)can be?xed theoretically.

From Chernysheva et al.[7],for the evaporator temperature un-der a certain heat load Q,T v;e¼T a

þ

1

a c;ext c;extþR wallþ

1

a c;int c;intþ

X

i

D P iÁ

dT

þ1

a e e;acti v e

!

ÁQ;

ðA:10Þwhere a c,ext is the convective heat transfer coef?cient at the external surface of the condenser and S c,ext is the total heat transfer surface

area of the?ns in the condenser;a c,int the heat transfer coef?cient during the vapor condensation in the condenser serpentine pipe and S c,int is the inner surface area of the condenser pipe;R wall is the thermal resistance of the condenser pipe wall;a e is the equivalent evaporation heat transfer coef?cient in the evaporator and S e,active is the evaporator surface area to which heat Q is sup-plied(active area);

P

i

D P i is the total pressure drop of vapor during the motion of vapor from the evaporator into the condenser and given by Eq.(A.3);R wall¼d wall=ðk wallÁS wallÞ,d wall is the thickness of the condenser tube and S wall is the outer surface area of the con-

denser tube;The derivative dT

dP

is a thermophysical characteristic of the working?uid which is taken along the liquid–vapor satura-tion line and can be calculated from the Clausius–Clapeyron equa-tion approximately at a reference temperature.It should be noted that a c,ext,a c,int and a e are very dif?cult to calculate accurately from any theoretical correlations(of course,in principle,the theoretical solutions for these coef?cient may exist)and more realistically these coef?cients should be identi?ed by careful experiments.For the prototype of the loop heat pipe in this study,all of these coef?-cient and parameters were presented in Refs.[5,6].

With the aid of Clausius–Clapeyron equation for water–vapor saturation lines,the?tting equation for water based on the pub-lished data is

P v;e¼exp

25:511À1:065Áh fg=R g

T e;vþ273:15

:ðA:11Þ

And the saturation pressure of vapor in the vapor removal chan-nels can be determined by substituting Eq.(A.10)into Eq.(A.11).

From Young–Laplace equation D P cap¼P v;eÀP l;w¼2r

r meniscus

,the li-quid pressure at the liquid–vapor interface in the wick can be ob-tained as,

P l;w¼P v;eÀD P cap:ðA:12ÞBy adding the pressure loss during the liquid?owing inside the wick,the saturated pressure inside the compensation chamber can be identi?ed as,

P l;cc¼P l;wþD P l;w:ðA:13ÞOnce more,by applying the equation of(A.11),the temperature of two phase mixture of water in the compensation chamber can be?xed!

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