Applied Energy 135 (2014) 586–596

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Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Thermal numerical model of a high temperature heat pipe heatexchanger under radiation

http://dx.doi.org/10.1016/j.apenergy.2014.08.0920306-2619/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +82 55 269 3630; fax: +82 55 269 3739.E-mail addresses: euiguk.jung@lge.com (E.G. Jung), jhboo@kau.ac.kr (J.H. Boo).

Eui Guk Jung a,⇑, Joon Hong Boo b

a SAC Laboratory, SAC Business Unit, AE Division, LG Electronics Inc., 76, Seongsan-dong, Changwon-city, Gyeongnam 641-713, Republic of Koreab School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang, Gyeonggi-do 412-791, Republic of Korea

h i g h l i g h t s

� Thermal modeling of the HPHEX for the high-temperature.� Radiation heat transfer analysis in the high-temperature HPHEX.� Prediction of the temperature distribution by adopting nodal approach.� Design and analysis of the HPHEX for the high-temperature.

a r t i c l e i n f o

Article history:Received 2 February 2014Received in revised form 5 August 2014Accepted 25 August 2014

Keywords:High-temperatureHeat pipe heat exchangerLiquid metal heat pipeSimulationEffectivenessRadiant heat transfer

a b s t r a c t

The heat transfer of an air-to-air heat pipe heat exchanger (HPHEX) with counter flow and a high-tem-perature range was modeled. The HPHEX was constructed from sodium-stainless steel (STS) heat pipes(HPs) using a staggered configuration. The thermal numerical model was developed by the nodalapproach, and the junction temperature and thermal resistance of the HP and heat transfer fluid of eachrow were defined. Surface-to-surface radiant heat transfer was applied to each row of the liquid metalHPHEX. The cold-side inlet air temperature was determined by iteration to converge to the minimumoperating temperature of the sodium HP. The cold-side inlet velocity and position of the common wallwere considered as the main variables in evaluating the performance of the liquid metal HPHEX, and theireffects on the temperature distribution, effectiveness, heat transfer rate of each row were investigated.The proposed row-by-row heat transfer model is useful for understanding the temperature distributionof each row and can be used to predict the cold-side inlet temperature of a liquid metal HPHEX withcounter flow. The recovery heat and effectiveness of the heat exchanger were calculated for various con-figurations and operating conditions. The simulation results agreed with experimental data to within 5%error for normal operation of the heat pipes, and within 11% error when the minimum temperature waslower than could allow normal operation of the sodium heat pipes.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Heat pipes (HPs) are two-phase heat transfer devices with highlyeffective heat transfer capabilities. Heat exchangers (HEXs) withHPs or thermosyphons (TSs) have superior heat transfer capabilitiescompared to conventional HEXs. Furthermore, they are useful forgenerating a high heat flux in a small space and are widely used inindustry because of their structural stability and economic advanta-ges [1–3]. The operating principle of HPs in which vaporization andcondensation are repeated in the evaporator and condenser isdescribed in detail in Refs. [4,5]. The working fluid inside the

evaporator vaporizes as it absorbs thermal energy. The vapor is thentransported to a condenser by the vapor pressure and then liquefiedby a cooling medium in the condenser. The liquid is returned to theevaporator by the capillary pressure produced by a capillarystructure. A HP can also be used over various temperature rangesdepending on the material of the container and the operatingtemperature of the working fluid.

Theoretical and experimental studies have been conducted toinvestigate medium- and low-temperature HPHEXs [1–3,5–7].These previous works addressed the working characteristics, ther-mal performance, and practical application of HPHEXs, as well astheories for their practical design. The equation of the optimumarea ratio between the hot-and cold-side in terms of the totalthermal resistance was presented in Ref. [5] and the optimum

Nomenclature

A area (m2)c specific heat (kJ/kg �C)D fin density (fins/m)d diameter (m)F view factorf friction factorG mass velocity (kg/s m2), G½¼ _m=rAfr �h heat transfer coefficient (W/m2 �C)H height (m)HP heat pipeHPHEX heat pipe heat exchangerJ radiosity (W/m2)j index for row number or Colburn factork thermal conductivity (W/m �C)L HPHEX length (m)_m mass flow rate (kg/s)

N number of heat pipes in a rowNTU number of transfer unitsp pipe or fin pitchP pressure (Pa)pr Prandtl numberQ heat transfer rate (W)R thermal resistance (�C/W)Re Reynolds numberr radial position (m) or radius (m)St Stanton numberS fin spacing (m)T temperature (�C)U overall heat transfer coefficient (W/m2 �C)u velocity (m/s)v specific volume (m3/kg)X tube bank pitch (m)Xd tube bank diagonal pitch, ðXT=2Þ2 þ X2

L

h i1=2

Greek symbolsd fin thickness (m)r ratio of free flow area to frontal area, Amin/Afr or Stefan–

Boltzmann constant (W/m2 K4)e effectiveness or emissivityg fin efficiency or overall extended surface efficiencyl fluid dynamic viscosity (m2/s)u porosity of capillary wick structure

q density (kg/m3)

Subscriptsa ambientc cold-sidecap screen capillary structurec cold-sideeff effectiveequ equivalentf fluidfr frontalfin finfin?in from fin to inletfin?out from fin to outletfin?p from fin to pipef-fin between fluid and finsh hydraulic or hot-sidein inletj row indexj ? j + 1 from jth row to (j + 1)th rowj ? j � 1 from jth row to (j � 1)th rowj ? j + 2 from jth row to (j + 1)th rowj ? j � 2 from jth row to (j � 2)th rowj + 1 ? j from (j + 1)th row to jth rowj � 1 ? j from (j � 1)th row to jth rowj + 2 ? j from (j + 2)th row to jth rowj � 2 ? j from (j � 2)th row to jth rowL lengthmean mean valuemin minimum valueo outer or overallout outletp pipe or pressurep ? fin from pipe to finp ? in from pipe to inletp ? out from pipe to outletrad radiationre recoveryT totalw wall or widthwi wick innerwf working fluid inside HP

E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596 587

effectiveness from the viewpoint of the operating limit of the HPswas considered in Ref. [6]. Soylemez [7] also presented aneconomic design methodology that considered the optimumeffectiveness and critical effectiveness.

Liquid metals such as sodium, lithium, potassium, and cesiumhave been used for high-temperature applications of over 500 �C,which require the container materials to be chemically inert [8]and resistant to high-temperature corrosion [9–11] by the workingfluid. Whereas a number of analytical and experimental studieshave been conducted on medium- and low-temperature HPHEXs,relatively few studies have been devoted to high-temperatureHPHEXs [12,13].

Yoo et al. [12] presented experimental results on the perfor-mance of an HPHEX that consisted of 90 sodium-stainless steel(STS) HPs in ten rows. A maximum recovery heat rate of 115 kWwas achieved by applying various hot- and cold-side inlet condi-tions. The experimental results suggested that the cold-side inletair temperature should be above 450 �C for normal operation ofthe HPs in all the rows and the achievement of satisfactory HPHEXperformance. Zhuang et al. [13] experimentally investigated the

performance of an HPHEX that used sodium as the working fluid,with hot- and cold-side inlet temperatures of 1000 �C and 20 �C,respectively, and an inlet pressure of 0.19 MPa on both sides.Previous studies have mainly developed theories on HPHEX perfor-mance based on the NTU-e relation [1–3,5,6]. Although the NTU-emethod is useful for predicting the temperature at the hot- andcold-side outlets, it is of limited usefulness in determining the walltemperature of the HPs and the temperature distribution of theheat transfer fluid in each row. Furthermore, there is currentlyno analytical model applicable to high-temperature liquid metalHPHEXs. Such a model should reflect the effect of radiation, whichcould be ignored in medium- and low-temperature ranges.

In this study, a thermal analysis model was developed by thenodal approach and used to predict the thermal performance of ahigh-temperature HPHEX. The numerical thermal model was usefulfor determining not only the temperature distribution of the fluid ineach row but also the temperatures at various points of the HP. Thiswas because the analytical outlet results for one row wereconsidered the inlet conditions for the next row. In particular, sur-face-to-surface radiant heat transfer was applied to each row for

XL

2aXT

do

di

bXd

Fig. 2. Unit cell of the layout of the staggered HPs with plain flat fins.

588 E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596

the heat transfer analysis of a HPHEX with a sodium HP and counterflow. The present study was aimed at using analytical methods todevelop a simulation tool that could be applied to the normal air-to-air operation of a sodium-STS HPHEX (or gas-to-air operation,considering that the hot-side medium is normally the combustionor exhaust gas). To ensure realistic results, the minimum tempera-ture of the sodium HP during normal operation was considered tobe 500 �C, which is based on the fact that the typical HP operatingtemperature range reported in literature is 400–1200 �C [4,5]. Inthe design of a liquid metal HPHEX with counter flow, the airtemperature at the cold-side inlet should be limited so that the walltemperature of the HP condenser positioned at the cold-side inletwould exceed the minimum operating temperature.

A lower air temperature at the cold-side inlet is advantageousfrom an economic point of view because more energy is requiredto heat the air as the air temperature at the cold-side inletincreases. If the air temperature at the cold-side inlet is reducedwhile the wall temperature of the HP condenser positioned atthe cold-side inlet is maintained at close to the minimum operat-ing temperature, better operating conditions from an economicpoint of view may be achieved. High-temperature corrosion isavoided by using STS as the pipe material as long as the operatingtemperature does not exceed 800 �C [11,14]. Using the air temper-ature at the cold-side inlet as a constraint, the air velocity at thecold-side inlet and the position of the common wall were used asthe main variables in the present study. The effects of thesevariables on the recovery heat, air temperature at the cold-sideinlet, effectiveness, and temperature distribution of each row wereinvestigated by row-by-row heat transfer analysis.

2. Thermal model of the heat pipe heat exchanger

Fig. 1 shows the theoretical simulation domain of the HPHEXwith counter flow; the temperature and velocity of each row areindicated. The heat transfer fluid on the hot-side flows into theinlet, and the HP supplies thermal energy to the heat transfer fluidon the cold-side. The excellent isothermal characteristics of theHPHEX make it unnecessary to subdivide the flows on the hot-and cold-sides into interspersed multiple flow passages. A simplermodeling method than that required to model a conventional heatexchanger was thus adopted.

Fig. 2 shows a unit of the staggered configuration of the HP andthe design variables are also indicated. The basic assumptionsapplied to the HPHEX analysis model and radiant heat transferwere described in detail in Refs [15,16,17], respectively. In additionto the above, the following assumptions are made:

1) The outer wall and tubing system constitute an adiabaticboundary, and convective and radiant thermal losses to theatmosphere were therefore not considered.

Tc,out

uc,out

Th,in

uh,in

Tc,in

uc,in

Th,out

uh,out

Tc,j

uc,j

Th,j

uh,j

Tc,j+1

uc,j+1

Th,j+1

uh,j+1

nj1 j+1

common wall

HHP

Hh

plain flat fin HP

H*(Hh/HHP )

Fig. 1. Calculation domain of the HPHEX.

2) To consider the view factors, each plain flat fin can be trea-ted as being equivalent to a circular fin with an adiabatic tip.

3) Thermal loss due to convection at the entrance and exit ofthe HPHEX is ignored, and only thermal loss due to radiationis considered.

4) Radiant heat transfer between the (jth) row of flat fins andthe (j + 1)th row of HPs are only considered for the next row.

2.1. Conductive and convective heat transfer modeling

Eqs. (1)–(3) express the geometric relationships regarding thearrangement of the HPs with fins attached on their hot-side. Asindicated by Eq. (1), the primary area of each row on the hot-andcold-sides is the sum of the areas of the HPs and the commonwalls.

Aj;p ¼pdoH�Nj;pðHHP � dfinDfinHHPÞ þ ðXLLW � pd2

ONj;p=4Þpdoð1� H�ÞNj;pðHHP � dfinDfinHHPÞ þ ðXLLW � pd2

ONj;p=4Þ

(

ð1Þ

where H⁄[= Hh/HHP] is the ratio of the evaporator length to the totallength of the HPs and Lw is width of the HPHEX.

The area of the fins in each row is given by

Aj;fin ¼2H� XLLW � pd2

o4

� �Nj;p

h iDfinHHP

2ð1� H�Þ XLLW � pd2o

4

� �Nj;p

h iDfinHHP

8><>: ð2Þ

The lower lines of Eqs. (1) and (2) correspond to the cold-side.The total heat transfer area of each row is given by

Aj;T ¼ Aj;p þ Aj;fin ð3Þ

The geometric configuration of the HPs is shown in Fig. 2. Thehot-and cold-sides have the same configuration, although there isflexibility for controlling the thermal performance by using a dif-ferent fin thickness or density for either side. The Reynolds numberof the flow through the HP arrangement is given by Eq. (4).

Rej ¼ qjuj;mdo=lj ð4Þ

where the velocity uj,m is obtained in the minimum free flow areadefined in Refs. [18,19]. Eq. (5) gives the coefficient of the convec-tive heat transfer between the air and the extended surface of theHP tube bank.

hj;o ¼ StjGcp;j ð5Þ

where the Stanton number Stj ¼ jj=Pr2=3j . Regarding the heat transfer

of plain flat and circular fins, various forms of the empirical equa-tion for jj are reported in Ref. [18] with respect to the geometricdimensions and the Reynolds number range.

E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596 589

Eq. (6) gives the overall efficiency in terms of the extended andtotal surface areas and the fin efficiency.

gj;o ¼ 1� Aj;fin

Aj;Tð1� gj;finÞ ð6Þ

The flat fin efficiency based on the equivalent flat fin height Hequ

was applied to the staggered tube bundle configuration in Ref.[20], resulting in the following:

gj;fin ¼tanðmjHequÞ

mjHequð7Þ

where the equivalent height of the plain flat fin Hequ [=XTXL/4do

� do/2] is determined by the dimensions of the staggered tubeconfiguration.

Fig. 3 shows the heat flow circuit of the hot-and cold-sides, andEqs. (8)–(10) give the thermal resistances at certain nodes of thehot fluid and HPs on the hot-side.

Rj;f -fin;h ¼1

gj;o;hhj;o;hAj;fin;hð8Þ

The thermal resistance of the wall of a HP is given by the following:

Rj;w;h ¼1

gj;o;hhj;o;hAj;p;hþ lnðdo=diÞ

2pkj;HP;hH�HHPNj;pð9Þ

The thermal resistance of the screen capillary structure saturatedwith the working fluid in the HP evaporator is given by Eq. (10).

Rj;cap;h ¼lnðdcap;o=dcap:iÞ

2pkj;eff ;hH�HHPNj;pð10Þ

where kj;eff ;h ¼kwf ;HP ½ðkwf ;HPþkcapÞ�ð1�/Þðkwf ;HP�kcapÞ�

kwf ;HPþkcapþð1�/Þðkwf ;HP�kcapÞ

n ois the effective thermal

conductivity of the working fluid and the screen capillary structureof porosity u[=1 � pSMdw/4]. S is the crimping factor and isassigned a typical value of 1.05 [4], and. M and dw are respectivelythe mesh number and wire diameter.

Eqs. (11)–(13) give the thermal resistances at certain nodes ofthe cold fluid and HPs on the cold-side. In particular, Eq. (11) givesthe thermal resistance of the screen capillary structure saturatedwith the working fluid in the HP condenser.

Rj;cap;c ¼lnðdcap;o=dcap;iÞ

2pkj;eff ;cð1� H�ÞHHPNj;pð11Þ

where kj,eff,c is the effective thermal conductivity of the workingfluid and screen capillary structure, and is the same as in Eq. (10).

The thermal resistance of the wall of a HP is given by thefollowing:

Rj;w;c ¼1

gj;o;chj;o;cAj;p;cþ lnðdo=diÞ

2pkj;HP;cð1� H�ÞHHPNj;pð12Þ

The thermal resistance between the fins and the cold fluid is given by

Rj;f -fin;c ¼1

gj;o;chj;o;cAj;fin;cð13Þ

Fig. 3. Thermal circuit of the HPHEX.

A TS does not include a capillary structure, and Eqs. (10) and (11)can thus be excluded from the analysis of a HPHEX that utilizes TSs.

The total thermal resistance of each row is given by

Rj;HP ¼ Rj;f -fin;h þ Rj;w;h þ Rj;cap;h þ Rj;cap;c þ Rj;w;c þ Rj;f -fin;c ð14Þ

Eq. (15) is an expression of the heat transfer rate of each row whenheat is transferred by convection and conduction through the HPsfrom the hot-side to the cold-side.

Qj;HP ¼ ðTj;f ;h � Tj;f ;cÞ=Rj;HP ð15Þ

2.2. Radiant heat transfer modeling

Fig. 3 is a schematic illustration of the flow of thermal energy,including by radiation, from the hot-side to the cold-side. It is nec-essary to consider radiant heat transfer in a thermal energyexchange process at temperatures greater than 600 �C [21]. Theeffects of the radiant heat transfer between the rows on the hot-and cold-sides are not considered because the two sides are sepa-rated by the common wall. The radiant heat transfer from the com-mon wall to the surfaces of the HP-fin assemblies on the hot- andcold-sides is not considered because the surface area of the commonwall is considerably smaller than that of each of the HP-fin assembly(the former is determined to be less than 0.1% of the latter).

By thin wall approximation, the surface temperature of an HP isconsidered to be identical to that of the wall, and the energy con-servation in the walls of the HPs in each row on the hot-side isexpressed by Eq. (16). As indicated by Eq. (16), the differencebetween the thermal energy transferred from the heat transferfluid to the HP-fin assembly by convection and conduction, andthe radiant heat energy (Qj,rad,T,h = Qj,p,rad,h + Qj,fin,rad,h) emitted fromthe wall of the HPs in a given row to the surface of the HP-finassembly of another row on the hot-side is equal to the differencebetween Qj,HP and the radiant heat energy (Qj,rad,T,c = Qj,p,rad,c +Qj,fin,rad,c) emitted from the walls of the HPs in a given row to thesurface of the HP-fin assembly of another row on the cold-side.Qj,rad,T,h and Qj,rad,T,c are the net radiant heat transfer rates of eachrow on the hot-and cold-sides of the HP-fin assembly.

ðTj;f ;h � Tj;w;hÞ=ðRj;f -fin;h þ Rj;w;hÞ � Q j;rad;T;h ¼ Q j;HP � Q j;rad;T;c ð16Þ

The energy conservation in the fins on the hot-side is expressed by

ðTj;f ;h � Tj;fin;hÞ=Rj;f -fin;h � Qj;rad;T;h ¼ Q j;HP � Qj;rad;T;c ð17Þ

The energy conservation in the walls of the HPs on the cold-side isexpressed by Eq. (18), which is derived similarly as Eq. (16).

Qj;HP � Q j;rad;T;c � ðTj;w;c � Tj;f ;cÞ=ðRj;w;c þ Rj;f -fin;cÞ ¼ 0 ð18Þ

The energy conservation in the fins on the cold-side is expressed by

Qj;HP � Q j;rad;T;c � ðTj;w;c � Tj;f ;cÞ=Rj;f -fin;c ¼ 0 ð19Þ

The temperatures of the HP walls and fins on the hot- and cold-sideare respectively given by Eqs. (20)–(23).

Tj;w;h ¼ Tj;f ;h � ðRj;f -fin;h þ Rj;w;hÞ½Qj;HP � ðQj;rad;c � Q j;rad;hÞ� ð20ÞTj;fin;h ¼ Tj;f ;h � Rj;f -fin;h½Q j;HP � ðQ j;rad;T;c � Q j;rad;T;hÞ� ð21ÞTj;w;c ¼ Tj;f ;c þ ðQ j;HP � Q j;rad;cÞðRj;f -fin;c þ Rj;w;cÞ ð22ÞTj;fin;c ¼ Tj;f ;c þ Rj;f -fin;cðQ j;HP � Q j;rad;cÞ ð23Þ

The surface radiation properties of gray and diffuse surfaces used invarious radiant heat transfer applications are often taken into con-sideration [16,17]. It is assumed that the absorptivity and emissivityof gray and diffuse surfaces have identical values. Radiation inten-sity can be easily expressed based on only the point on a surfaceat which the emission and reflection of radiation are diffuse. Theparameter is thus widely used in engineering applications. As

1 2 j n-1 n

Fp in,1

Fp in

Fp in,2Ffin in,1

Fp, 1 2

Fp, 1 2

Fp, j j-1

Fp, j j-2

Fp, j j-2

Fp, n n-2

Fp, n n-2

Fp, 2 j+2

Fp, j j-1

Fp, j j+1 Fp, n-1 n

Fp out,1

Fp out,1

Fp out,n-1

Ffin out,1

Ffin p,j

outletinlet

Fig. 4. Surface-to-surface radiation process.

590 E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596

shown in Fig. 4, the staggered tube bundle, including the plain flatfin, has a very complex geometric shape. Moreover, the radiant heatexchanges between tube and tube and between tube and fin shouldbe taken into consideration. The surface of a single row canexchange radiant heat with the surfaces of several other rows. Con-sidering that radiant heat exchange between surfaces can be ahighly complex process, simplification is thus required. As shownin Fig. 4, five rows are used as a standard unit of the radiant heatexchange, and the radiant heat exchange between the flat finsand the HPs are only considered for the next row. The surface ofeach row of the HP-fin assemblies on the hot- and cold-sides isconsidered to be gray, and the wall diffuse. The total radiant heattransfer between the surfaces is given by Eq. (24). Qj,rad,p is the radi-ant heat transfer rate between HP and HP, and Qj,rad,fin is the radiantheat transfer rate between HP and fin.

Q j;rad;T ¼ Q j;rad;p þ Q j;rad;fin

¼ e1� e

rT4j;w � Jj;w

� �Aj;p þ

e1� e

rT4j;fin � Jj;fin

� �Aj;fin ð24Þ

where Jj,w and Jj,fin are the radiosities of the HPs wall and fin surface.The radiocities (Jj,w and Jj,fin) of each row and the summationPkk¼1Jk;wFj!k

� �of the radiosities between surfaces, according to

row of reciprocity, is given in Appendix A. As shown in Fig. 4, a sin-gle HP in the jth row can exchange radiant thermal energy with twoHPs in each of the (j + 1)th and (j � 1)th rows, and with a single HPsurface in each of the (j + 2)th and (j � 2)th rows. The first terms inthe parentheses in the square brackets in Eqs. (A.1) and (A.2), rep-resents the rate of radiant thermal loss through the inlet (hot-side)or outlet (cold-side) from the surfaces of the HPs and fins in thefirst row and second row, respectively. The first terms in the paren-theses in the square brackets in Eqs. (A.4) and (A.5) represent therate of radiant thermal loss through the inlet (cold-side) or outlet(hot-side) from the surfaces of the HPs and fins in the (n � 1)th

and nth rows, respectively. The radiosity between the jth fin andthe (j + 1)th HP in each row is given by Eq. (A.6).

The values of the radiosities obtained by Eqs. A.1, A.1.1, A.2,A.2.2, A.3, A.3.1, A.4, A.4.1, A.5, A.5.1, A.6 can be expressed inmatrix form as in Eq. (25).

A1 0 0 0 0 00 A2 0 0 0 0

..

. ... ..

. ... ..

. ...

0 0 Aj 0 0 0

..

. ... ..

. ... ..

. ...

0 0 0 0 An�1 00 0 0 0 0 An

266666666666664

377777777777775

J1;w

J2;w

..

.

Jj;w

..

.

Jn�1;w

Jn;w

266666666666664

377777777777775¼

C1

C2

..

.

Cj

..

.

Cn�1

Cn

266666666666664

377777777777775

ð25Þ

Each term of the matrix is summarized in Appendix B. The radi-osities of each row can be obtained using Eq. (25), which can besolved by the Gauss–Seidel iteration method. The view factorsbetween two cylinders of finite and infinite lengths, respectively,have been investigated [21,22]. Although the equation of the viewfactors between two tubes of infinite lengths was also thoroughlyconsidered in Ref. [21], it has limited practical application owing tothe very large errors in its results [22]. Thus, in Ref. [22], the viewfactors Fp between two cylinders of finite lengths are obtained byexponential interpolation of the results of numerical analyses.

Perrotin and Clodic [23] used different forms of the equivalentradius to approximate rectangular and hexagonal fins to circularfins. As shown in Fig. 4, a plain flat fin in a bundle of tubes in a stag-gered tube configuration can be approximated to a hexagonal fin.The view factors between the fins and the tube were obtained bythe equivalent circular radius (requ[= 0.635XT(Xd/XT � 0.3)]) approx-imation. The equations of the view factors Ffin between the tubes andcircular fins and the equations of those (Ffin?in, Ffin?out, Fp?in, Fp?out)of the rows exposed to the inlet and outlet were specificallydetermined in Ref. [21].

The air temperatures at the hot-and cold-side outlets are givenby Eqs. (26) and (27), respectively.

Tjþ1;h ¼ Tj;h �Tj;f ;h � Tj;w;h

ð _mhcp;hÞjðRj;f -fin;h þ Rj;w;hÞ

¼ Tj;h � ðTj;f ;h � Tj;f ;cÞ=ðRj;HP _mj;hcp;j;hÞ ð26Þ

Tj;c ¼ Tjþ1;c þTj;w;c � Tj;f ;c

ð _mccp;cÞjðRj;f -fin;c þ Rj;w;cÞ

¼ Tjþ1;c þ ðTj;f ;h � Tj;f ;cÞ=ðRj;HP _mj;ccp;j;cÞ ð27Þ

3. Solution procedure

The air temperature at the cold-side inlet under the counterflow is obtained by assuming the air temperature at the cold-sideoutlet of the first row. Eqs. (16)–(25) require numerical solutionsbecause the temperatures at different points of the HPHEX areunknown. The flow chart applied to the calculation is presentedin Fig. 5. The radiant heat transfer rate is set to zero in the initialstage. Then, by assuming the air temperature at the inlet of thecold-side, the temperatures at different points in each row of theHPHEX can be precisely obtained. After calculating the radiositiesby substituting the determined temperature of each node in Eq.(25) and Eq. (A.6), the total radiant heat transfer rate can beobtained using Eq. (24). The four energy conservative equations,Eqs. (16)–(19), have to satisfy the total radiant heat transfer rateof the hot-and cold-sides calculated by Eq. (24). The final solutioncan be obtained by increasing or decreasing air temperature of the

START

Geometricaldimension of HP and

HPHEX

Inlet conditionTin,h, uin,h, Tin,c, uin,c

heat transfer areaAj,fin,h, Aj,p,h, Aj,fin,c,

Aj,p,c

heat transfer coeffiecientand fin efficiency

hj,o, j,o, j,fin

thermal resistanceRj,f-fin,h, Rj,w,h, Rj,cap,h

Rj,f-fin,h, Rj,w,h, Rj,cap,h, Rj,HP

Qj,HP, Tj,w,h, Tj,fin,h, Tj,w,c, Tj,fin,c

assumptions:(1) no radiation: Qj,rad,T,h= 0, Qj,rad,T,c= 0(2) outlet temperature of the cold side: Tout,c

Qj,rad,p,h, Qj,rad,fin,h, Qj,rad,p,c, Qj,rad,fin,c, Qj,rad,T,h,Qj,rad,T,C

Eq. (18), (19), (20) and 21

NoQj,rad,T,h, Qj,rad,T,c,Tout,c

Tn,w,c = 500°C±0.1%

NoTout,c

yes

Qj,HP, Qj,rad,T,h,Qj,rad,T,c,Qre,Qre, T,Tj,w,h,

Tj,fin,h, Tj,w,c, Tj,fin,c

STOP

yes

view factorFp, Ffin, Ffin in,Ffin out

Fp in, Fp out

radicitiesEq. (25)

Fig. 5. Flow chart of numerical simulation.

E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596 591

cold-side outlet until the energy equations of the hot-and cold-sides (Eqs. (16)–(19)) are satisfied, with an error margin of ±0.1%.Furthermore, the exact air temperature at the cold-side inlet ofthe nth row can be obtained by iterative calculation until theobtained air temperature at the cold-side inlet is within a certainerror range (±0.1%) of the target temperature (500 �C) of an HP wallon the cold-side inlet. For a liquid metal HPHEX, it is necessary todetermine the air temperature at the cold-side inlet that makes the

temperature of the HP condenser positioned at that inlet close tothe minimum operating temperature. The properties of the airare temperature-dependent and are applied to the calculation byexponential fitting as functions of the temperature of the heattransfer fluid.

In the NTU-e method used in previous studies, the totaleffectiveness of a system with multiple rows was determined fromthe effectiveness of the hot-and cold-sides of each row, and the air

-1 0 1 2 3 4 5 6 7 8 9 10 110

200

400

600

800

1000

1200

1400

hot gas volume flow rate = 16 m3/scold Air Volume Flow Rte = 16.8 m3/s

numerical experimental [12]hot gas: heat pipe: cold air:

Tem

pera

ture

, o C

Row number, j

Fig. 7. Comparison of the theoretical and experimental results for HPs with averagewall temperature below 500 �C (H⁄ = 0.5).

0.6 0.8 1.0 1.2 1.4 1.650

60

70

80

90

100

110

120

uc,in

[m/s]

Qre [

kW]

experimental results [12] numerical results

0

10

20

30

40

50

60

U [W

/m2]

experimental results [12] numerical results

Fig. 8. Comparison of the numerical and experimental results for recovery heat andoverall thermal coefficient (H⁄ = 0.5).

592 E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596

temperatures at the hot-and cold-side outlets were then deter-mined from the total effectiveness [1,2,6,7]. In the nodal approachto heat transfer analysis used in this study, the temperatures atthe hot- and cold-side outlets of each row were obtained fromthe thermal resistance. Thus, the effectiveness of each row of thehot-and cold-sides was not required.

4. Results and discussion

4.1. Model validation

The reliability of the thermal model was verified by the exper-imental results in Ref. [12], as shown in Figs. 6–8. Unfortunately,experimental results for a conventional high-temperature HPHEXare currently unavailable in international journals. The reliabilityof the model was therefore experimentally verified using a domes-tic conference paper as reference. Typical comparisons betweenthe simulation results of this study and the experimental resultsin the referenced paper for a sodium-STS HPHEX are shown inFigs. 6 and 7. The experimental heat pipe temperatures in the fig-ures were determined by averaging the temperatures of the wallsof the HP evaporator and condenser. The ratio between the hot-and cold-side heat transfer areas, H⁄, in the figures is 0.5, whichwas the value used for the experiment. The detailed specificationsand inlet conditions of the experimental model of the sodium-STHPHEX are available in reference.

Fig. 6 summarizes the results for the 1st to 10th rows, where allthe heat pipe temperatures are above 500 �C. The theoretical resultsagree well with those of the experiment to within 5%. The inlet tem-peratures and volumetric flow rates of the hot -and cold-sides are820 �C, 210 �C, 16 nm3/s, and 16.8 nm3/s, respectively.

Fig. 7 shows a typical case of when the heat pipe operating tem-perature at the cold-side inlet is below 500 �C. In the actual opera-tion of the heat pipe, the operating temperature of the cold-sideinlet is 380 �C when the inlet gas temperatures and volumetric flowrates of the hot-and cold-sides are 1000 �C, 210 �C, 10.7 nm3/s, and16.8 nm3/s, respectively. The maximum error in the cold-side airtemperature is 12% (which occurs in the 5th row) and that in theHP operating temperature is 10.2%.

Fig. 8 shows the recovery heat and overall heat transfer coeffi-cient (U[= Qre/AT(Tc,out � Tc,in)]), which were obtained by changingthe cold-side velocity under the following conditions: the inlettemperatures of the hot- and cold-sides were 800 �C and 380 �C,respectively, and the air speed on the hot-side was set to 1.33 m/s. In general, the recovery heat investigated relatively well below

0

100

200

300

400

500

600

700

800

900

1000

hot gas volume flow rate = 16 m3/s

cold air volume flow rte = 16.8 m3/s

numerical experimental [12]hot gas: heat pipe: cold air:

Tem

pera

ture

, o C

-1 0 1 2 3 4 5 6 7 8 9 10 11

Row number, j

Fig. 6. Comparison of the numerical and experimental results for HPs with averagewall temperature above 500 �C (H⁄ = 0.5).

a relative error of 5%, whereas the overall heat transfer coefficientwas found to be somewhat larger, with a relative error of 11%.

4.2. Effects of the design parameter

Figs. 9–15 show the analytical results for the liquid metalHPHEX, and Table 1 gives the detailed design parameters and thebasic input conditions. The HPHEX utilizes sodium-STS HPs witha staggered arrangement. Each odd row has ten HPs and each evenrow has nine. There are 20 rows altogether, and counter flow of thehot-and cold fluids is employed. As shown in Fig. 1, the 1st row isthe hot-side inlet, and the 20th row is the cold-side inlet. For con-venience, the temperature and velocity of the air at the hot-sideinlet were fixed at 800 �C (to avoid high-temperature corrosionof the stainless steel) and 3 m/s (Reh,in = 1213), respectively. Theair temperature at the cold-side inlet was determined such thatthe wall temperature of the HP condenser at the cold-side inletwould be close to the minimum operating temperature (±0.1%).

Fig. 9 shows the air temperatures at the hot- and cold-side out-lets with respect to Re⁄ for various values of H⁄. As indicated byEqs. (1) and (2), the heat transfer area of the hot-side increases withincreasing H⁄, whereas the heat transfer area of the cold-sidedecreases. As shown in Fig. 9, the maximum value of Re⁄ that

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

440

460

480

500

520

540

560

580

600

620

640

hot-side cold-side

H* = 0.3 H

* = 0.3

H* = 0.5 H

* = 0.5

H* = 0.7 H

* = 0.7

Out

let t

empe

ratu

re [

o C]

Re*

Fig. 9. Hot-and cold-side outlet air temperatures as functions of H⁄.

160

200

240

280

320

360

400

440

480

Tc,in

H* = 0.3

H* = 0.5

H* = 0.7C

old

side

inle

t tem

pera

ture

[o C

]

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

Re*

Fig. 10. Cold-side inlet air temperature as a function of H⁄.

0.0 0.4 0.8 1.2 1.6 2.0 2.40.30

0.35

0.40

0.45

0.50

0.55

0.60

H* = 0.3

H* = 0.5

H* = 0.7

Re*

Eff

ectiv

enes

s

Fig. 11. Effectiveness as a function as H⁄.

25

30

35

40

45

50

55

60

65

Qre

H* = 0.3:

H* = 0.5:

H* = 0.7:

Qre

[kW

]

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

Re*

Fig. 12. Total recovery heat as a function of H⁄.

2

3

4

5

6

7

QT,h,rad

QT,c,rad

H* = 0.3:

H* = 0.5:

H* = 0.7:

Tot

al r

adia

nt h

eat t

rans

fer

rate

[kW

]

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

Re*

Fig. 13. Total radiant heat transfer rate as a function of H⁄.

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

QT,h,rad,loss

QT,c,rad,loss

H* = 0.3:

H* = 0.5:

H* = 0.7:

Tot

al r

adia

nt th

erm

al lo

ss [

kW]

Re*

Fig. 14. Total radiant thermal losses on hot-and cold-sides as functions of H⁄.

E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596 593

satisfies the minimum operating temperature decreases withincreasing H⁄, being 2.67, 1.32, and 0.76 for H⁄ values of 0.3, 0.5,and 0.7, respectively. The air temperature at the hot-side outletincreases with increasing H⁄, whereas that at the cold-side outletdecreases. For Re⁄ = 0.6 and H⁄ = 0.3, the temperatures at the

hot- and cold-side outlets were respectively 595.9 and 492 �C, andwere determined to change to 630.5 and 446 �C when H⁄was chan-ged to 0.7. Furthermore, when H⁄ was 0.7, the increase in Re⁄

resulted in a decrease in the cold-side outlet temperature. As shownin Fig. 9, the increase in Re⁄was accompanied by a rapid decrease inthe cold-side inlet temperature, as well as a decrease in the cold-side outlet temperature.

2 4 6 8 10 12 14 16 18 20500

525

550

575

600

625

650

675

700H

* = 0.5 radiation no radiaton

Re* = 1.31:

Re* = 0.56:

Re* = 0.29:

T w,c [

o C]

Row number, j

Fig. 15. Comparison of cold-side wall temperature distributions with and withoutconsideration of radiation (H⁄ = 0.5).

594 E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596

Fig. 10 shows the air temperature at the cold-side inlet withrespect to Re⁄ for various values of H⁄. To obtain realistic results,the minimum temperature of the sodium HPs for normal operationwas considered to be 500 �C in this study, based on typical HPoperating temperature range of 500–1200 �C. In the design of aliquid metal HPHEX with counter flow, the air temperature at thecold-side inlet should be limited such that the wall temperatureof the HP condenser at the cold-side inlet would exceed the mini-mum temperature. The air temperature at the cold-side inlet forwhich the HP wall temperature at the cold-side inlet is minimumdecreases with increasing H⁄, and increases with decreasing Re⁄.For Re⁄ = 0.7, the cold-side inlet air temperature was determinedto be 444.5 and 194.2 �C for H⁄ values of 0.3 and 0.7, respectively.

Fig. 11 shows the total effectiveness with respect to Re⁄ forvarious values of H⁄. The effectiveness is given by Eq. (28)

eT ¼_mccp;cðTc;out � Tc;inÞð _mcpÞminðTh;in � Tc;inÞ

¼_mhcp;hðTh;in � Th;outÞð _mcpÞminðTh;in � Tc;inÞ

ð28Þ

The minimum effectiveness was observed to satisfy the condi-tion _mhcp;h ¼ _mccp;c (see Eq. (28). As H⁄ increased, the minimumeffectiveness occurred at a lower value of Re⁄. As shown in Figs. 9and 10, the air temperatures at the cold-side inlet and outlet bothdecreased with increasing H⁄, causing the air temperature at thehot-side outlet to increase. Accordingly, for Re⁄ values below0.63, the effectiveness reduced with increasing H⁄. Overall, thetotal effectiveness tends to increase with decreasing height of thehot-side, Hh, and increasing height of the cold-side.

Fig. 12 shows the total recovery heat as a function of Re⁄ for var-ious values of H⁄. The Recovery heat was determined using Eq. (29).

Table 1Design parameters and basic input conditions of the high-temperature HPHEX.

Item Dimensions/material/input condition

Heat exchanger 1 m (H) � 0.5 m (W) � 1.01 m (L), stainless steelHeat pipe do = 25.4 mm, di = 22.1 mm, L = 1 m, stainless stee

mesh number: 200, 2 layersPlain flat fins Thickness: 1 mm, fin density: 270.3 fins/m, pitch:Arrangement of heat pipes XT = 50.5 mm, XL = 50.5 mmNumber of heat pipes per row Odd row: 10, even row: 9Total number of rows 20Heat exchanger fluid Cold-side: air, hot-side: gas (regarded as air)Hot-side inlet temperature and

velocity800 �C, 3 m/s (Reh,in = 1213)

Flow type Counter flow

Qre ¼Xn

j¼1

Q HP;j ¼ _mcðcp;outTc;out � cp;inTc;inÞ ð29Þ

As shown in Figs. 9 and 10, an increase in H⁄ is accompanied bya decrease in the outlet and inlet temperatures on the cold-side.However, the temperature difference between the inlet and outleton the cold-side increases with increasing H⁄. Hence, the value ofthe recovery heat also increases with increasing H⁄. For H⁄ = 0.3,the maximum and minimum values of the temperature differencebetween the inlet and outlet of the cold-side are respectively 146.9and 22.9 �C, whereas they are 262.8 and 105.1 �C for H⁄ = 0.7.Furthermore, for H⁄ = 0.3, the maximum and minimum values ofthe recovery heat are respectively 31.7 and 26.5 kW, whereas theyare 62.83 and 56.14 kW for H⁄ = 0.7.

Fig. 13 shows the total radiant heat transfer rate on the hot- andcold-sides as a function of H⁄. The total radiant heat transfer rate isthe sum of the radiant heat transfer rates of all the rows on thehot-and cold-sides. The inlet temperatures on the hot-side werefixed for all cases. As also shown in Fig. 9, an increase in H⁄ producedan increase in the outlet temperature, which in turn increased thenet radiant heat transfer rate on the hot-side. Conversely, the netradiant heat transfer rate on the cold-side was decreased as theincrease in H⁄ reduced the wall temperature of the HPs on thecold-side and the heat transfer fluid (see Figs. 9 and 10). Further-more, the radiant heat transfer rate on both sides also increased withincreasing Re⁄. This was because the increase in Re⁄ was accompa-nied by an increase in the temperature distribution of the HPHEX.

Fig. 14 shows the summation of the radiant heat losses from theinlet and outlet of the hot- and cold-sides to the surrounding as afunction of Re⁄ for various values of H⁄. In the case of the hot-side,the outlet temperature increases with increasing H⁄ (see Fig. 9),and the radiant thermal loss therefore increases; the maximumradiant heat loss is 0.91 kW (H⁄ = 0.7). Regarding the cold-side,the temperature of the heat transfer fluid decreases with increas-ing H⁄, and the radiant heat loss therefore decreases; the maximumradiant heat loss is 0.79 kW (H⁄ = 0.3). Generally, the radiant heatloss on both sides is less than 1 kW.

Radiant heat transfer is often ignored in high-temperature heattransfer analysis. When radiant heat transfer is taken into consider-ation in a HPHEX, iterative calculations should be used to determinethe wall temperature of the HPs. This process requires considerabletime and effort. Fig. 15 compares the cold-side wall temperature dis-tributions of the HPHEX of this study with and without consider-ation of radiation, as a function of Re⁄. Generally, when radiantheat transfer is considered, the heat transfer rate increases and thetemperature distribution becomes more uniform. Moreover, thetemperatures are higher when radiation is considered. The HP walltemperatures on the hot-and cold-sides are respectively given byEqs. (20) and (22). When radiation was not considered, the maxi-mum error in the cold-side HP wall temperature was determinedto be approximately 5.2% for Re⁄ = 1.31, with the error increasing

316l 316 L (e = 0.95), working fluid: sodium, screen capillary structure: STS 316L,

3.7 mm, spacing: 2.7 mm

E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596 595

with increasing temperature. The no-radiation assumption wasobserved to be reasonably valid only when a maximum relativeerror of 5.2% was permitted. However, caution is required for higheroperating temperatures.

5. Conclusions

A thermal model of a liquid metal HPHEX with counter flowwas developed using the nodal approach. The thermal resistanceof the plain flat fins was determined by treating them as circularfins with an adiabatic tip and using the equivalent circular radius.The temperatures of the HP walls and the air temperature distribu-tions on the cold- and hot-sides were obtained for each row of HPs.It was established that the air temperature at the cold-side inlet ofa liquid metal HPHEX with counter flow should be limited suchthat the temperature of the HP condenser at the cold-side inletwould be the minimum operating temperature, which is deter-mined by iterative calculations.

The developed simulation process was validated by comparisonwith experimental results. Good agreement (within 5% relativeerror) was observed when the operating temperatures of all theheat pipes of the HHPX were higher than 500 �C, which was theminimum temperature necessary for full operation of the sodiumheat pipes. This condition was especially important for the rowof HPs at the cold-side air inlet. However, the error increased toas much as 10.2% when any of the heat pipes operated below theminimum temperature.

An increase in H⁄ narrowed the range of Re⁄ values that could sat-isfy the minimum operating temperature of the HPs, although it alsoreduced the inlet temperature and pressure drop on the cold-side.The effectiveness of the HPHEX was strongly dependent on H⁄ andRe⁄, the values of which could be adjusted to the required operatingcondition. Thus, in the design and fabrication of a liquid metalHPHEX, a suitable position of the common wall and value of Re⁄

should be selected to enhance the effectiveness. Conditions that willreduce the cold-side air inlet temperature are also advantageousfrom an economic point of view. Comparison of the calculationresults obtained by considering and without considering radiationrevealed a maximum relative error of 5.2%. However, consideringthat the error increased with increasing temperature, caution isrequired in ignoring radiation for high operating temperatures.

Appendix A

Radiosities in each row and the summation of the radiositiesbetween surfaces, according to law of the reciprocity

1st row

Jj;w ¼ erT4j;w þ ð1� eÞ r T4

j;wFj;p;p!in þ T4j;finFfin;fin!in

� �þXk

k¼1

Jk;wFj!k

" #

ðA:1Þ

Xk

k¼1

Jk;wFj!k ¼ er 2 T4j�1;wFj!j�1;p þ T4

jþ1;wFj!jþ1;p

� �hn

þT4jþ2;wFj!jþ2;p þ T4

j;finFp!fin;fin

iþð1� eÞJj;w 2ðFj!j�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ

�þFj!jþ2;pFjþ2!j;p þ Fp!fin;finFfin!p;fin

��ðA:1:1Þ

2th row

Jj;w ¼ erT4j;w þ ð1� eÞ

Xk

k¼1

Jk;wFj!k ðA:2Þ

Xk

k¼1

Jk;wFj!k ¼ er 2 T4j�1;wFj!j�1;p þ T4

jþ1;wFj!jþ1;p

� �þ T4

jþ2;wFj!jþ2;p

hn

þT4j;finFp!fin;fin

iþ ð1� eÞJj;w 2ðFj!j�1;pFj�1!j;p

�þFj!jþ1;pFjþ1!j;pÞ þ Fj!jþ2;pFjþ2!j;p þ Fp!fin;finFfin!p;fin

��ðA:2:2Þ

jth row

Jj;w ¼ erT4j;w þ ð1� eÞ

Xk

k¼1

Jk;wFj!k ðA:3Þ

Xk

k¼1

Jk;wFj!k ¼ er 2 T4j�1;wFj!j�1;p þ T4

jþ1;wFj!jþ1;p

� �hn

þT4jþ2;wFj!jþ2;p þ T4

j�2;wFj!j�2;p þ T4j;f�finFp!fin;fin

iþð1� eÞJj;w 2ðFj!j�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ

�þFj!jþ2;pFjþ2!j;p þ Fj!j�2;pFj�2!j;p þ Fp!fin;finFfin!p;fin

��ðA:3:1Þ

(n � 1)th row

Jj;w¼ erT4j;wþð1�eÞ r T4

j;wFj;p!out;pþT4j;finFj;fin!out;fin

� �þXk

k¼1

Jk;wFj!k

" #

ðA:4Þ

Xk

k¼1

Jk;wFj!k ¼ er 2 T4j�1;wFj!j�1;p þ T4

jþ1;wFj!jþ1;p

� �hh

þT4w;j�2Fj!j�2;p þ T4

j;finFp!fin;fin

iþð1� eÞJn�1;w 2ðFj!j�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ

�þFj!j�2;pFj�2!j;p þ Fp!fin;finFfin!p;fin

��ðA:4:1Þ

nth row

Jj;w¼ erT4j;wþð1�eÞ r T4

j;wFj;p!out;pþT4j;finFj;fin!out;fin

� �þXk

k¼1

Jk;wFj!k

" #

ðA:5Þ

Xk

k¼1

Jk;wFj!k ¼ er 2T4j�1;wFj!j�1;p þ T4

j�2;wFj!j�2;p þ T4j;f -finFp!fin;fin

� �hþð1� eÞJn;w 2Fj�1!j;pFj!j�1;p þ Fj�2!j;pFj!j�2;p

�þFfin!p;finFp!fin;fin

��ðA:5:1Þ

between fins and HPs in each row

Jj;fin ¼ erT4j;fin þ ð1� eÞ

X1

k¼1

Jk;finFj!k ðA:6Þ

Xk

k¼1

Jk;finFj!k ¼ erT4jþ1;wFfin!p;fin þ ð1� eÞJj;finFp!fin;finFfin!p;fin ðA:6:1Þ

Appendix B

Equations of the matrix elements (Eq. (25)) of the high-temperature HPHEX

A1 ¼ 1� ð1� eÞ2ð2Fjþ1!j;pFj!jþ1;p þ Fjþ2!j;pFj!jþ2;p þ Ffin!p;finFp!fin;finÞðB:1Þ

596 E.G. Jung, J.H. Boo / Applied Energy 135 (2014) 586–596

A2 ¼ 1� ð1� eÞ2 2ðFj!�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ�

þFj!jþ2;pFjþ2!j;p þ Fp!fin;finFfin!p;fin�

ðB:2Þ

Aj ¼ 1� ð1� eÞ2 2ðFj!j�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ�

þFj!jþ2;pFjþ2!j;p þ Fj!�2;pFj�2!j;p þ Fp!fin;finFfin!p;fin�

ðB:3Þ

An�1 ¼ 1� ð1� eÞ2 2ðFj!j�1;pFj�1!j;p þ Fj!jþ1;pFjþ1!j;pÞ�

þFj!j�2;pFj�2!j;p þ Fp!fin;finFfin!p;fin�

ðB:4Þ

An ¼ 1� ð1� eÞ2½ð2Fj�1!j;pFj!j�1;p þ Fj�2!j;pFj!j�2;p

þ Ffin!p;finFp!fin;finÞ� ðB:5Þ

C1 ¼ erT4j;w þ ð1� eÞr T4

j;wFj;p!in;p þ T4j;finFj;fin!in;fin

� �hþe 2T4

jþ1;wFj!jþ1;p þ T4jþ2;wFj!jþ2;p þ T4

fin;jFfin;p!fin

� �iðB:6Þ

Cj ¼ erT4j;w þ ð1� eÞr e 2 T4

j�1;wFj!j�1;p þ T4jþ1;wFj!jþ1;p

� �hnþT4

jþ2;wFj!jþ2;p þ T4j�2;wFj!j�2;p þ T4

j;finFp!fin;fin

ioðB:7Þ

C2 ¼ erT4j;w þ ð1� eÞr T4

j;wFj;p!in;p þ T4j;finFj;fin!in;fin

� �nþe 2 T4

j�1;w;Fj!j�1;p þ T4jþ1;wFj!jþ1;p

� �þ T4

jþ2;wFj!jþ2;p þ T4j;finFp!fin;fin

h ioðB:8Þ

Cn�1 ¼ erT4j;w þ ð1� eÞr T4

j;wFj;p!out;p þ T4j;finFj;fin!out;fin

� �nþe 2 T4

j�1;wFj!j�1;p þ T4jþ1;wFj!jþ1;p

� �þ T4

w;j�2Fj!�2;p þ T4j;finFp!fin;fin

h ioðB:9Þ

Cn ¼ erT4j;w þ ð1� eÞr T4

j;wFj;p!out;p þ T4j;finFj;fin!out;fin

� �hþe 2T4

j�1;wFj!j�1;p þ T4j�2;wFj!j�2;p þ T4

j;finFp!fin;fin

� �iðB:10Þ

References

[1] Silverstein CC. Design and technology of heat pipes for cooling and heatexchanger. 1st ed. London: Taylor & Francis, Inc.; 1992. Chapter 3–5.

[2] Noie SH. Investigation of thermal performance of an air-to-air thermosyphonheat exchanger using e-NTU method. Appl Therm Eng 2006;26:559–67.

[3] Yang F, Yuan X, Lin G. Waste heat recovery using heat pipe heat exchanger forheating automobile using exhaust gas. Appl Therm Eng 2003;23:367–72.

[4] Peterson GP. An introduction to heat pipes. 1st ed. Singapore: John Wiley &Sons, Inc.; 1994. Chapter 1 and 5.

[5] Faghri A. Heat pipe science and technology. 1st ed. London: Taylor & Francis,Inc.; 1995. Chapter 1–11.

[6] Abd El-Baky MA, Mohamed MM. Heat pipe heat exchanger for heat recovery inair conditioning. Appl Therm Eng 2007;27:795–801.

[7] Soylemez MS. On the thermoeconomical optimization of heat pipe heatexchanger HPHE for waste heat recovery. Energy Convers Manage2003;44:2509–17.

[8] Yamamoto T, Tanaka Y. Long-term operation test of sodium heat pipe. In:Proceedings of 5th international heat pipe conference. Japan; 1984. p. 126–32.

[9] Tu ST, Zhang H, Zhou WW. Corrosion failures of high temperature heat pipes.Eng Fail Anal 1999;6:363–70.

[10] Jang JH. Startup characteristics of a potassium heat pipe from the frozen state. JThermophys Heat Transfer 1995;9:117–22.

[11] Davis DR. ASM specialty handbook – stainless steels. ASM Int 1994.[12] Yoo JH, Kang HK, Lee YI, Park SW, Lee DC, Yoo JR. An experimental study on the

operational characteristics and performance of the sodium heat pipe heatexchanger. In: Proceedings of the society of air-conditioning and refrigeratingengineers of Korea summer annular conference; 2004. p. 725–30.

[13] Zhuang J, Yang RW. Development of liquid metal heat pipe heat exchanger. In:Proceedings of 9th international heat pipe conference part I; 1995. p. 7–17.

[14] Park SY, Boo JH. An experimental study on the heat transfer characteristics ofhigh temperature cylindrical heat pipes. Korean J Air-Cond Refrig Eng2004;16:70–6.

[15] Jung EG, Boo JH. Thermal analytical model of latent thermal storage with heatpipe heat exchanger for concentrated solar power. Sol Energy2014;102:318–32.

[16] Carotenuto A, Ruocco G, Reale F. Heat exchange in a multi-cavity volumetricsolar receiver. Sol Energy 1991;46(4):241–8.

[17] Jung EG, Boo JH, Kang YH, Kim NH. Effectiveness of a multi-channel volumetricair receiver for a solar power tower. Heat Mass Transf 2013;49:1181–90.

[18] Shah RK, Sekalic DP. Fundamentals of heat exchanger design. 1st ed. NewYork: Wiley & Sons, Inc.; 2003. Chapter 5–8.

[19] Incropera FP, Dewitt DP. Fundamental of heat and mass transfer. 3rded. Toronto: John Wiley & Sons, Inc.; 2002. Chapter 3–5.

[20] Song KW, Wang Y, Zhang Q, Wang LB, Liu YJ. Numerical study of the finefficiency and a modified fin efficiency formula for flat tube bank fin heatexchanger. Int J Heat Mass Transf 2011;54:2661–72.

[21] Modest MF. Radiative heat transfer. 1st ed. Singapore: McGRAW-HILL, Inc.;1993. Chapter 3–5 and view factor catalogue.

[22] Bopche SB, Sridharan A. Determination of view factors by contour integraltechnique. Ann Nucl Energy 2009;36:1681–8.

[23] Perrotin P, Clodic D. Fin efficiency calculation in enhanced fin-and-tube heatexchanger in dry conditions. In: Proceedings of international congress ofrefrigeration 2003. Washington, D.C., ICR0026.