1-s2.0-S1290072918312419-main

International Journal of Thermal Sciences 137 (2019) 276–287

ContentslistsavailableatScienceDirect

InternationalJournalofThermalSciences

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

Experimentalandnumericalinvestigationofforcedconvectionheattransfer

inporouslatticestructuresproducedbyselectivelasermelting

T

J.Y.Ho a ,K.C.Leong a,∗ ,T.N.Wong b

a SingaporeCentrefor3DPrinting,SchoolofMechanicalandAerospaceEngineering,NanyangTechnologicalUniversity,50NanyangAvenue,Singapore,639798,Republic

ofSingapore

b SchoolofMechanicalandAerospaceEngineering,NanyangTechnologicalUniversity,50NanyangAvenue,Singapore,639798,RepublicofSingapore

ARTICLEINFO

Keywords:

Single-phase

Forcedconvection

Latticestructures

Porousmedia

Selectivelasermelting

ABSTRACT

Forced convection heattransfer in four structured porousmaterials produced by selectivelasermelting (SLM)

was experimentally and numerically studied. The porous materials are lattice structures consisting of periodic

arrangementsofRhombi-Octetunitcells.Thelatticestructureshavesimilarporosity(ε)butareofdifferentunit

cellsizesof5mm,7mm,10mmand12mm.Thisinvestigationaimstocharacterizeandevaluatethethermohydraulicpropertiesofthisnewclassoflatticestructures.Thehydrodynamicandheattransfercharacteristicsof

the lattices structures such as the permeability (K), inertia coefficient (C E ) and Nusselt number (Nu) were determinedexperimentallyinanairflowchannelinwhichtheReynoldsnumber(Re)canbevariedbetween1300

and7000.Usingalinearheatconductionsetup,thestagnanteffectivethermalconductivities(k eff )ofthevarious

lattice structures were determined and a relationship between k eff and the ligament width (d) of the lattice

structure was obtained. Based on the local thermal non-equilibrium model, numerical simulations were performed

to determine the interfacial heat transfer coefficients (h sf ) of the lattice structures. Our results showed

thatthepressuredrop(ΔP)andNusseltnumber(Nu)ofthelatticestructuresincreasewithdecreasingdandthe

highestNuof906wasobtainedwiththeL1latticestructurewhichhasthesmallestligamentwidth.Thelattice

structures alsoexhibited high k eff values which were up to5.5timeshigher thanthatofthe aluminum foams.

Duetotheorderlyarrangementsofthelatticestructures,theirpermeability-basedfrictionfactors(f⋅Da 1/2 )were

foundtobelowerthanthemetallicfoams.However,theirh sf valueswerealsolower.TheColburnj-factorand

thermalefficiencyindex(η)ofthelatticestructuresweredeterminedandfoundtobehigherthanthoseofthe

commercialmetallicfoamsandconventionalpinfinheatsinks.Insummary,thisinvestigationdemonstratesthe

promisinguseofanewclassoflatticestructures(Rhombi-Octet)forenhancingsingle-phaseforcedconvection

cooling.

1. Introduction

Metallic foams have demonstrated high potential in enhancing

forcedconvectionheattransfer[1,2].Theseopencellfoamshavelarge

surfacearea-to-volumeratiosandhighporosities,arelightandconsist

ofintertwiningporousnetworkswhichinducetortuousfluidpathsand

promotefluidmixing.Thesekeyfeatureshaverenderedthemsuitable

for convectivecoolingapplications suchas in electronicthermalmanagement

devices and compact heat transfer devices in the aerospace,

automobileanddefenseindustries[3,4].

Duetothesereasons,theenhancementofsingle-phaseheattransfer

using metallic foams has been extensively investigated. For instance,

Boomsmaetal.[5]demonstratedthatcompressedmetallicfoamscould

reduce the thermal resistance of a heat exchanger by almost half as

comparedtoplateheatexchangers.BhattacharyaandMahajan[6]investigatednaturalconvectioninaluminummetallicfoamheatsinksfor

application in electronic cooling. It was reported that the metallic

foams with finned structure enhanced heat transfer rates more significantlyascomparedtotheblockmetallicfoamsandtheheattransfer

coefficients were found to increase with increasing fin density. Leong

and Jin [7,8] conducted oscillating and steady flow experiments

through a channel filled with aluminum foams of different pore densities.

Under oscillating flow condition, it was determined that the

average Nusselt number increases with increasing pore density and

kineticReynoldsnumber.Inaddition,thealuminumfoamsunderboth

oscillating and steady flow also exhibited higher heat transfer performance

than conventional finned heat sinks. Hatami and Ganji [9]

theoretically investigated the thermal performance of circular porous

∗ Correspondingauthor.

E-mailaddress:mkcleong@ntu.edu.sg(K.C.Leong).

https://doi.org/10.1016/j.ijthermalsci.2018.11.022

Received13July2018;Receivedinrevisedform9November2018;Accepted20November2018

Available online 02 December 2018

1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.

J.Y.Hoetal. International Journal of Thermal Sciences 137 (2019) 276–287

Nomenclature

A basearea(m 2 )

A s , t totalheattransferarea(m 2 )

C E inertiacoefficient

d ligamentwidthordiameter(m)

d p porediameter(m)

Da Darcynumber

D h hydraulicdiameter(m)

f modifiedfrictionfactor–Eq.(11)

f 1 frictionfactor–Eq.(30)

H heightoflatticestructure(m)

h¯ averageheattransfercoefficient(W/m 2 ⋅K)

h sf interfacialheattransfercoefficient(W/m 2 ⋅K)

j Colburnj-factor

k bulkthermalconductivity(W/m·K)

k eff stagnanteffectivethermalconductivity(W/m·K)

K permeability(m 2 )

l ligamentlength(m)

L length(m)

Nu NusseltnumberbasedonflowchannelD h

Nu d Nusseltnumberbasedond

q'' heatflux(W/m 2 )

Q heatrate(W)

Re ReynoldsnumberbasedonflowchannelD h

Re d Reynoldsnumberbasedond

St Stantonnumber

T temperature(°C)

U averageairvelocity(m/s)

V volume(m 3 )

GreekSymbols

ε porosity

ρ density(kg/m 3 )

μ dynamicviscosity(kg/m⋅s)

η efficiencyindex

Subscripts

c coldside

h hotside

f fluid

s solid

w wall

in inlet

finswithdifferentsectionshapesandmaterialssubjectedtoconvection

and radiation heat transfer. Using the least square method, the fin

equations for the porous fins were solved to obtain the most exact

analyticalsolution.DogonchiandGanji[10]subsequentlyextendedthe

study to a moving fin where the Differential Transformation Method

(DTM) was employed to solve the fin equation analytically. Their results

show that the fin temperature distribution decreases with increasing

Peclet number and radiation-conduction parameter. CylindricalpipesfilledwithmetalfoamswereanalyticallystudiedbyLuetal.

[11].Bysolvingthelocalthermalnon-equilibriumheattransfermodel

and the Brinkman-extended Darcy momentum model for the porous

media, it was concluded that the effects of foam thermal conductivity

werelesssignificantatlowReynoldsnumbers.Inaddition,forfoamsof

lowthermalconductivities,varyingtheporedensityalsohadlittleinfluence

on the heat transfer performance. These findings suggest that

low thermal conductivity and highly porous foams can be used to reduce

cost and pressure drop. Despite demonstrating improvements in

convective cooling, metallic foams typically suffer from low effective

thermalconductivitiesofbetween2and7W/m·K[12].Inaddition,the

poorinterconnectivityoftheirinternalporestructureshasalsoresulted

inhighresistancetofluidflowwhichlimitstheirpracticalapplications

suchasinplate-finheatexchangers[13].

In the recent years, periodic lattice structures have emerged as

multi-functionalmaterialswiththepotentialforbothloadbearingand

heat transfer. In comparison with the stochastic network of metallic

foams, periodic lattice structures consist of orderly unit cell arrangements.

The unit cells are usually an assembly of cylindrical struts. By

varying the strut arrangements, different unit cell configurations such

asKagomé,honeycombandpyramidalstructurescanbeproduced[14].

Owingtotheuniformityofthelatticestructureunitcellarrangements,

high strength and stiffness at a given weight can be achieved. Furthermore,

the strut diameter and length of the unit cell can also be

adjusted to produce a homogenous three-dimensional network with

enhanced thermal and hydrodynamic performances. Investigations on

thelatticestructures’mechanicalpropertiesofvarioustopologieshave

beenreportedbyLiuetal.[15]andHyunetal.[16].Acomprehensive

review of the thermal and hydrodynamic characteristics of periodic

latticestructuresisprovidedbyLuetal.[17].

The convection heat transfer of air in lattice structures of the tetrahedral

unit cell was investigated by Kim et al. [18]. Due to the

anisotropicityoftheunitcell,thepressuredropsvariedunderdifferent

unit cell orientations. However, heat transfer rates were almost independent

of the unit cell orientation. It was determined that the formation

of horseshoe vortices and flow field-to-lattice structure interactions

at the vertices of the unit cell increased the overall

enhancements in Nusselt number. Subsequently, Tian et al. [19] experimentally

characterized square and diamond-shaped periodic cellularstructures

madefrom wire-screen laminatesand determined that

theoverallthermalefficiencyindices(η)ofthecellularstructureswere

approximately 3 times larger than stochastic copper metallic foams.

Thesehigher ηvaluesofthecellularstructureswerelargelyduetotheir

lowerflowresistances.Theuseofthehollowmicro-latticestructureina

cross-flow heat exchanger was demonstrated by Maloney et al. [20].

The micro-lattice was fabricated by electroplating nickel onto a sacrificial

polymer micro-lattice. Experiments were performed by running

hotwater through theinternalflow channels ofthe micro-latticewith

cold water flowing over its external surfaces. The overall thermal

conductances per unit volume of the heat exchanger were determined

to be between 0.84W/cm 3·K and 1.58W/cm 3·K and a correlation was

developedtopredictthethermalperformancesoftheheatexchangerat

differentlengthscales.Morerecently,Sonetal.[21]conductedasystematic

study on the forced convection heat transfer of air in lattice

structuresofdifferentporosities.Thetetrahedralunitcellwasusedand

the porosity was varied by using struts of different diameters. The

Nusseltnumbersofthelatticestructureswereobservedtoincreasewith

decreasingporosityandthefrictionfactorswerefoundtobeanorderof

magnitude smaller than those of the metallic foams of similar porosities.Itwasalsoconcludedthatthelatticestructureshavesimilarheat

transferperformanceasthemetallicfoamsbutwithsignificantlylower

flowresistances.

Fromtheabovereview,itcanbeseenthatperiodiclatticestructures

have superior thermal and hydraulic characteristics and their performances

are mainly dependent on the solid heat conduction paths and

the fluid-solid interactions. These mechanisms are in turn affected by

the lattice structure parameters such as the unit cell topology, strut

diameter and porosity. However, likely due to the limitation of conventional

manufacturing techniques, existing unit cell designs of the

lattice structures are largely limited to simple geometries such as

square, diamond or pyramidal shapes. To further enhance the lattice

structures thermal and hydraulic performances, new unit cell

277


geometries with high effective thermal conductivities and low flow

resistances can be developed. Lattice structures of different strut diameters

and porosities should also be produced so that parametric studiescanbeconductedtounderstandtheireffectsonthefluidflowfield

andtoobtaintheperformancematricesforcoolingapplications.

These challenges can be overcome by the recent advancements in

additive manufacturingtechniques.Selectivelasermeltingis abranch

ofadditivemanufacturing(AM)whichusesahigh-powerlasertomelt

and fuse the metallic base powder based on a predefined computeraideddesign.Complexthree-dimensionaldesignscanbe

fabricatedby

melting consecutive layers of powder over each other. The main advantage

of SLM over other AM technologies is its ability to produce

metallic components that are more than 99% dense for heat transfer

applications. Recently, SLM has been explored to fabricate novel

structures forsingle-phaseforcedconvection[22,23],nucleateboiling

[24,25] and condensation [26] studies where significant heat transfer

enhancementshavebeenreported.Inaddition,anewtypeofperiodic

lattice structure was also investigated by Ho et al. [27] for enhancing

the thermal and hydraulic performances of water-cooled cold plates.

The lattice structure, produced using SLM, consists of orderly arrangements

of the Rhombi-Octet unit cell and was employed as the

enhanced feature in a cylindrical flow channel. Their results showed

that the lattice structure exhibited up to 383% enhancement in the

averageheattransfercoefficient(h¯)ascomparedtoanemptychannel

and up to 51% higher h ave values as compared to the best performing

coldplatewithtwistedtapeinsert.Furthermore,inarecentstudyusing

waterascoolant,itwasalsodeterminedthatthenewlatticestructure

demonstrated higher h ave values than the conventional metallic foam

whilehavingsimilarpressuredrop[28].

AsacontinuationofapreviousstudybyHoetal.[27],thepresent

investigation focuses on characterizing the forced convection heat

transferandpressuredropacrossthelatticestructuresofRhombi-Octet

unitcellsusingairasthecoolingmedium.TheRhombi-Octetstructure

isselectedforthisinvestigationasitcanachievehighpackingdensity

withitsligamentarrangements.Inaddition,theperiodicarrangements

of the unit cells have shown potential in enhancing forced convection

heat transfer [27]. However, in the previous work [27], studies were

only performed to investigate the potential application of lattice

structuresasinsertsinthewater-cooledcoldplates.Inordertoachieve

abetter understandingoftheforcedconvectionheattransfer mechanisms

of this new type of lattice structure, a fundamental study to determine

its thermo-hydraulic properties such as permeability, inertia

coefficient, effective and interfacial thermal conductivities with differentunitcellsizesneedstobecarriedout.Inthisinvestigation,four

latticeheatsinksmadefromrepetitionsoftheRhombi-Octetunitcellof

the same porosity but of different ligament widths and surface-to-volume

ratios were fabricated. Experiments were conducted in a horizontal

flow channel to determine their permeabilities, inertia coefficientsandNusseltnumbers.Inaddition,theeffectivestagnantthermal

conductivitiesoftheselatticestructureswerealsomeasured.Usingthe

experimental data and numerical simulation, the interfacial heat

transfercoefficientsbetweenthefluidandsolidphaseswereevaluated

and the possible fluid-solid interactions were elucidated. Finally, the

thermal and hydraulic performances of these lattice structures were

compared against other enhanced structures and metallic open cell

foams. To the best of the authors’ knowledge, there is no published

work onthe fundamentalstudies offorcedconvectionheattransfer in

lattice structures of the Rhombi-Octet unit cell. Therefore, it is hoped

that from the present investigation, new insights on the transport mechanisms

can be obtained and the data collected can be used for the

developmentofenhancedcoolingdeviceswiththisnewclassoflattice

structures.

2. Fabricationandcharacterizationofspecimens

The SLM 280 (SLM Solutions GmbH) facility at the Future of

Manufacturing Laboratory 1 of the Singapore Centre for 3D Printing

(SC3DP) in Nanyang Technological University (NTU), Singapore was

employedtofabricatethespecimensusedinthepresentinvestigation.

The machine which consists of a Gaussian distributed Yb:YAG laser

Fig.1.(a)IsometricdrawingsofaRhombi-Octetunitcell,(b)comparisonofdifferentunitcellsizesand(c)projectedviewandvariousporetypesofaunitcell.

278


withmaximumpowerof400Wandlaserbeamspotsizeof80μmwas

utilizedtomeltandfusethebasemetallicpowder.Analuminumalloy

powder,AlSi10Mg,of20μm–63μmsizewasusedasthebasemetallic

powder to produce the test specimens for the experimental investigation.Aluminum

alloywasselectedasit hashigh thermalconductivity

of between 103W/m·K and 190W/m·K and is light as compared to

otheralloys.Thelasermeltingprocesswascarriedoutinthemachine's

buildchamberwhereinertargongaswasfirstusedtoflushthechamber

toattainanoxygenleveloflessthan0.2%soastominimizeoxidation

and combustion of powder. Subsequently, the first layer of AlSi10Mg

metallicpowderwasdistributedevenlyonthebase-platebyarecoater

and the laser beam was directed to melt the powder based on a preprogrammed

model. Upon completion of the laser melting process for

thefirstlayer,thebase-platewasthenloweredbyone-layerthicknessof

50μm and the process was repeated until the parts are fully constructed.Inthepresentinvestigation,alaserpowerof350W,scanning

speed of 1150mm/s and hatching spacing, which is the spacing between

adjacent laser scanning tracks, of 0.17mm were used in the

fabricationbasedonpriorsuccessesofbuildingdensebulkobjects.

Four lattice heat sinks (L1 – L4) with the bulk dimensions of

40mm×90mm×15mmweredesignedandfabricatedfortheforced

convection heat transfer experiments. Each lattice structure is integratedontoabaseplateofdimensions40mm×90mm×3.5mmas

a single built piece and consists of periodic arrangements of the

Rhombi-Octetunitcells.ThedrawingofaunitcellisshowninFig.1(a)

andacomparisonofvariousunitcellsizesisshowninFig.1(b).Each

unitcellconsistsof18squareand24triangularfacesandexamplesof

the square and triangular faces are highlighted in blue and red in

Fig. 1(a), respectively. Each square is positioned at 45° to its adjacent

squares to form a Rhombicuboctahedron core whereasat each corner,

the triangles are arranged to form an Octet truss-like structure. A detaileddescriptionoftheunitcellgeometrycanbefoundinRef.[27].In

this study, the lattice heat sinks were made of repetitions of 5mm,

7mm, 10mm and 12mm unit cells. All the unit cells have the same

porosity(ε)of0.85butduetothedifferenceintheirunitcellsize,their

ligament length (l), ligament width (d), average pore diameter (d p,ave )

and surface area-to-volume ratio (A st /V) are different. The ligament

length(l)representsthestrutlengthonthesquarefaceoftheunitcell

andtheligamentwidth(d)isthethicknessofeachstrut.Repetitionsof

unit cells were arranged adjacent to each other to obtain the bulk dimensions

of 40mm×90mm×15mm of the heat sinks. However, in

somecases,theunitcellscouldnotbefittedintothebulkdimensionsof

the heat sinkand this resulted inthe truncation ofthese unit cells.As

theportionofunitcellbeingtruncateddiffersfordifferentunitcellsize,

this resulted the difference in porosity among the different heatsinks,

asshowninTable1.TheprojectedviewoftheRhombi-Octetunitcellis

showninFig.1(c).Itcanbeseenthateachunitcellconsistsofsixtypes

ofpores(d p,1 –d p,6 ).Inordertocomputed p,ave ,thehydraulicdiameter

of each pore was determined and d p,ave was computed by taking the

area-weighted average of all the pores using Eq. (1). The geometrical

parametersofalltheunitcellsaresummarizedinTable1.Imagesofthe

fourlatticeheatsinksareshowninFig.2(a)–(d).Thelatticeheatsinks

are named L1, L2, L3 and L4 and they are made of repetitions of

Rhombi-Octet unit cells of sizes 5mm, 7mm, 10mm and 12mm, respectively.

In addition to the lattice heat sinks, a heat sink with cylindricalpinfinarray(Fig.3(a))andaheatsinkwithparabolicpinfin

array(Fig.3(b))werealsofabricatedforcomparison.Thespecimensof

cylindricalpinfinandparabolicpinfinarraysarenamedasC1andP1,

respectively. The pin fins are in staggered arrangements. The cylindricalpinfinshaveadiameterof3.24mm,heightof15mmandlateral

and longitudinal fin-to-fin spacings of 4mm and 10mm, respectively.

On the other hand, the parabolic pin fins have a base diameter of

5.65mmwhichreduces to0.9mmatthefintip.Similartothecylindrical

pin fin array, the height of the parabolic fins is 15mm while its

lateral and longitudinal fin-to-fin spacings are 4mm and 10mm, respectively.

d

p,

ave

=

n d

n A

A

i p,

i i

i

i

The forced convection heat transfer performances of the lattice

structuresaresignificantlyaffectedbytheirstagnanteffectivethermal

conductivities (k eff ). Therefore, the k eff values of the lattice structures

were determined experimentally. For the thermal conductivity measurements,

another four specimens with the same lattice structures as

L1,L2,L3andL4werefabricated.ThesefourspecimensarenamedLc1,

Lc2, Lc3 and Lc4 and have unit cell sizes of 5mm, 7mm, 10mm and

12mm,respectively.ImagesofthespecimensareshowninFig.4.They

have bulk dimensions of 33.25mm×34.55mm×38mm. Each specimenconsistsofatopfacesheetof9mmheight,abottomfacesheetof

14mm height and a lattice structure of 15mm height. The lattice

structureissandwichedbetweenthetopandbottomfacesheets.Asthe

facesheetsandlatticestructurewerefabricatedasonebuiltpiece,there

is no thermal contact resistance at the face sheet and lattice structure

interface.Finally,asolidAl6061-T6blockandasolidAlSi10Mgblock

of the same bulk dimensions (33.25mm×34.55mm×38mm) were

fabricated and their thermal conductivities were also determined for

comparison.

3. Experimentalsetupandprocedures

3.1. Thermalconductivitytestfacility

Fig. 5 shows a schematic of the experimental facility used to determinek

eff ofthelatticestructures.Themeasurementswereperformed

on specimens Lc1, Lc2, Lc3 and Lc4. A copper block fitted with three

cartridge heaters is located above the top face sheet of the test specimen.

The cartridge heaters are connected to a variable transformer

whichallowstheheatratefromtheheaterstobecontrolled.Usingan

ammeter and a voltmeter, the current (I) and voltage input (V) to the

heatersweredeterminedandtheoutputheatrates(Q)fromtheheaters

were computed. A heat sink is located below the bottom face sheet

wheretheheatisdissipated.Tapwaterwasusedasthecoolingmedium

andwasallowedtorunthroughtheheatsinkflowchannels.Tominimizeheatlosses,Teflonisusedastheinsulatingmaterialanditencloses

thecopper,heatersandthetestspecimen.Alayerofelastomericfoamis

usedasasecondlayerofinsulationandisappliedontheoutersurface

oftheTeflon.AsdepictedinFig.6(a),eightK-typethermocouples(T h,1

–T h,8 )arefittedintothetopfacesheetofthetestspecimenat6.5mm

abovethetopfacesheetandlatticestructureinterface(Δy 1 ).Asshown

inFig.6(b),fourK-typethermocouples(T c,1 –T c,4 )areinstalledatthe

bottom face sheet at 4mm below the bottom face sheet and lattice

structureinterface(Δy 2 ).Inordertodeterminek eff ofthevariouslattice

structures,theone-dimensionalFourier'slawasshowninEq.(2a)was

firstusedtocomputethetemperatureatthetopfacesheetandlattice

structure interface (T top ), where T h is the average of T h,1 – T h,8 and

k AlSi10Mg isthethermalconductivityofbulkAlSi10Mg.Usingthesame

approach,thetemperatureatthebottomfacesheetandlatticestructure

interface(T bottom )wasobtainedusingEq.(2b),whereT c istheaverage

of T c,1 – T c,4 . Finally, with Eq. (2c), k eff of the lattice structure can be

determined. Throughout the experiments, six additional K-type thermocouples

were installed on the surface of the elastomeric foam to

estimate the heat loss to the surroundings. Based on the temperatures

recorded from these thermocouples and the correlations for natural

Table1

Geometricalparametersofthelatticestructureheatsinks.

Unitcellsize(mm) ε d(mm) l(mm) d/l d p,ave (mm) A st /V(m −1 )

5 0.8402 0.42 1.88 0.22 1.00 1355

7 0.8428 0.59 2.62 0.22 1.22 968

10 0.8458 0.84 3.76 0.22 1.54 678

12 0.8538 0.99 4.49 0.22 1.75 565

(1)

279


Fig.2.Imagesoflatticeheatsinks(a)L1,(b)L2,(c)L3and(d)L4usedforforcedconvectionheattransferinvestigation.

Fig.3.Imagesof(a)cylindricalpinfinheatsink–C1and(b)parabolicpinfin

heatsink–P1.

convectionprovidedinRef.[29],therateofheatlosswasdetermined

tobelessthan2.7%.UsingthemethodbyMoffat[30],theuncertainty

of k eff can be computed by Eq. (3). In this experimental setup, the accuracies

of the ammeter and voltmeter are 0.01 A and 1.0V, respectively

whereas the thermocouples were calibrated to an accuracy

of±0.5°C. Using Eq. (3), the maximum measurement uncertainty of

k eff wasfoundtobe±3.5%.

Priortothestartoftheexperimentswiththelatticestructures,the

accuracyof thesetup wasvalidatedusingasolidaluminum (Al6061-

T6)block.Fromtheheatratesoftheheaters(Q),cross-sectionalareaof

the aluminum block (A) and the temperature gradient ( dT

dy ) computed

fromthethermocouplemeasurements,thethermalconductivityofbulk

Al-6061(k Al )wasfoundtobe164W/m⋅Kwhichisclosetothevalueof

167W/m⋅K reported in Ref. [31], indicating good accuracy of the experimental

facility. Using the same procedures, the thermal conductivityofthebulkAlSi10Mg(k

AlSi10Mg )wasdeterminedtobe153W/

m⋅K.

Fig. 5.Schematic of experimental facility for thermal conductivity measurements.

T

top

=

8

T

i=

1 h, i ( Q/ A)

y1

8

k

AlSi10 Mg

(2a)

T

k

bottom

eff

4

T

i=

1 c, i ( Q/ A)

y2

= +

4 kAlSi10 Mg

(2b)

( Q/ A)

=

( T T )/ y

top bottom (2c)

dk

2 2 2 2 2 2

eff dI dV dA d y

dTtop

dTbottom

= + + + + +

keff

I V A y Ttop

Tbottom

Ttop

Tbottom

(3)

Fig.6.(a)Topview oftopfacesheetthermocouple positions,and (b)bottom

viewofbottomfacesheetthermocouplepositions.

3.2. Forcedconvectionheattransfertestfacility

ThetestfacilityshownschematicallyinFig.7wasemployedforthe

forced convection heat transfer investigation. This setup is similar to

thatofLeongetal.[32]andconsistsofaTeflonairflowchannelandan

Fig.4.Specimensfork eff measurements(a)Lc1–5mmunitcell,(b)Lc2–7mmunitcell,(c)Lc3–10mmunitcell,and(d)Lc4–12mmunitcell.

280


acrylic cover. Air supplied by an auto-balance compressor entered the

inlet contraction section of 9:1 contraction ratio. Subsequently, it was

channeled into a straight section where the various lattice structures

and pin fin heat sinks were placed. Two wire meshes of 0.32mm

aperture size which served as the air flowstraighteners were installed

alongthestraightsectionoftheflowchannelandupstreamofthetest

specimens.Ananemometerwaspositionedbetweenthewiremeshesto

measure the average air flow velocity (U) and a K-type thermocouple

was placed after the second wire mesh to measure the inlet air temperature

(T in ). A layer of elastomeric foam, which served as an additionallayerofinsulation,wasadheredontotheexternalsurfacesofthe

Teflonflowchanneltominimizeheatlosstothesurrounding.Thetest

sectionshowninFig.8(a)consistsofacopperblockandfourcartridge

heaters.Thecopperblockhasdimensionsof90mm×40mm×6mm

and each cartridge heater has a maximum heat rate of 60W and a

diameter of 6.35mm. The cartridge heaters were connected to a variable

transformer which enabled the output heat rate (Q) from the

heaters to be varied. Prior to the conduct of the experiments, the test

specimens were adhered onto the copper block using thermally conductiveepoxy.AsshowninFig.8(b),nineK-typethermocouples(T

w,1

–T w,9 )wereinsertedintothebaseplateoftestspecimenstoobtainthe

wall temperatures. Each thermocouple was positioned at 10mm apart

andwaslocatedat1.5mmbelowthebaseplatetopsurface.UsingEqs.

(4)–(7),theaverageheattransfercoefficient(h¯),Nusseltnumber(Nu)

andReynoldsnumber(Re)canbedetermined.Itshouldbenotedthat

inEq.(5),AdenotestheprojectedareaofthebaseplateandinEqs.(6)

and (7), D h is the hydraulic diameter of the flow channel. An inclined

manometer, connected to two pressure taps installed directly before

and after the test specimens, was used to measure the pressure drop

(ΔP) across the test specimens at different air velocities. Finally, eight

K-type thermocouples were also placed on the surface of the elastomericfoam.Usingthetemperaturesrecordedfromthesethermocouples

andthecorrelationsfornaturalconvectionofRef.[29],therateofheat

losstothesurroundingswasestimatedtobelessthan1.3%.Usingthe

methodbyMoffat[30],theuncertaintyofNucanbecomputedbyEq.

(8). The maximum measurement uncertainty of Nu was found to

be±5.6%.

T

w

=

9

T

i=

1 w,

i

9

h¯

Q/

A

=

( T T )

w in (5)

= hD ¯

h

Nu

kf (6)

(4)

f

Re = UD h

µ

f (7)

dNu

dI dV dA dTw

= + + + +

Nu I V A T T

4. Resultsanddiscussions

4.1. Pressuredrop

2 2 2 2 2

w

in

dTin

T T

Fig. 9showsthe measuredpressure dropper unitlength (ΔP/L) of

the lattice structures at different air velocities (U). It can be seen that

ΔP/L decreases with increasing unit cell size or increasing ligament

width(d)ofthelatticestructure.Forexample,atU=3.4m/s,ΔP/Lof

the L1 lattice structure is 29.3kPa/m whereas those of the L2 and L3

lattice structures are 21.8kPa/m and 13.4kPa/m, respectively. This

comparison shows that the increase in d from L1 to L2 resulted in a

reductioninΔP/Lof34.4%whereasincreasingdfromL2toL3resulted

inafurtherreductioninΔP/Lof62.7%.However,furtherincreaseofd

fromL3toL4resultedinalesssignificantreductioninΔP/L.Ascanbe

seenfromTable1,theincreaseindalsoresultedintheincreaseinthe

average pore diameter (d p,ave ). Due to a larger d p,ave , the resistance to

thefluidflowpathreducesandthereforeresultedinlowerΔP/Lvalues.

ThisexplainsthesignificantreductioninΔP/LoftheL2andL3lattice

structuresascomparedtoL1.However,theincreaseofd p,ave fromL3to

L4producedalesssignificantreductioninΔP/L.Thisislikelyduetothe

largerligamentwidth(d)ofL4eventhoughitsd p,ave valueislargerthan

L3. The large d value of L4 may have resulted in a higher form drag

whichproducedasmallerreductioninΔP/L.

The pressure drop across a porous medium under non-Darcy flow

conditionscanbeexpressedasaquadraticfunctionofU.Alsoknownas

theForchheimer-extendedDarcyequation,theexpressionshownasEq.

(9)relatesΔPtothepermeability(K)andinertiacoefficient(C E )ofthe

porous medium. While K depicts the ability of a porous medium to

transmit fluid, C E accounts for the effects of form drag and is strongly

dependent on the unit cell design and arrangement of the porous

medium. By employing the porous medium analogy on the lattice

structures,curvefittingoftheresultsofFig.9intheformofquadratic

functions was performed. By comparing the coefficients of the curvefittedresultswithEq.(9),KandC

E ofthevariouslatticestructureswere

determined and shown in Table 2. Using the Darcy number (Da) expressionofEq.(10),Da

1/2 wasalsocomputedandshowninTable2.As

expected,theKvaluesincreasewithincreasingligamentwidth(d)and

increasing average pore diameter (d p,ave ). For the C E values, the reductioninC

E wasobservedastheunitcellsizeincreases,i.e.,fromL1to

w

in

(8)

Fig.7.Schematicoftestfacilityforforcedconvectionheattransferinvestigation.

281


Fig.8.(a)Schematicoftestsection,and(b)positionsofthermocouples(T w, 1 –T w, 9 )inthebaseplateofaheatsink.

f Da

1/2

=

P

L

f

K

U

2

(12)

f Da

1/2

1

= + CE

ReDa1/2 (13)

4.2. Effectivethermalconductivity

Fig.9.Comparisonofpressuredropperunitlength(ΔP/L)ofvariousRhombi-

OctetlatticestructuresL1–L4.

L3.However,furtherincreaseintheunitcellsizefromL3toL4resulted

in a slight increase in the C E value. As explained above, this larger C E

value of L4 has resulted in a higher form drag as compared to L3 and

couldbeduetothelargerligamentwidthoftheL4structure.

The modified friction factor (f) which relates the pressure drop

across the lattice structures and the flow channel hydraulic diameter

(D h ) can be expressed as Eq. (11). For comparison against different

types of porous media, permeability (K) is commonly used as the

characteristiclengthandthiscanbeachievedbyusingtheproductoff

andDa 1/2 asshowninEq.(12).BymultiplyingD h /ρ f·U 2 onbothsidesof

Eq.(9),anexpressionwhichrelatesf⋅Da 1/2 ,Re⋅Da 1/2 andC E asshown

inEq.(13)canbederived.UsingtheexperimentallyobtainedK,C E and

Da 1/2 valuesofthevariouslatticestructures,asshowninTable2,the

f⋅Da 1/2 values at different Re⋅Da 1/2 were computed and presented in

Fig.10.Fornickel(Ni)foams,BeaversandSparrow[33]showedthata

C E valueof0.74providedagoodapproximationwhereasforaluminum

(Al)foams,Kim etal.[34]proposedaC E valueof0.1.ByusingtheK

values of 0.352×10 −7 m 2 and 0.51×10 −7 m 2 for one of the foam

configurations reported in Refs. [33,34], respectively and their corresponding

C E values, the f⋅Da 1/2 values for the Ni and Al foams are

plottedinFig.10.Likelyduetotheorderlyunitcellarrangementofthe

lattice structures, lower C E values were achieved with the lattice

structures as compared to the Ni and Al foams. This resulted in lower

frictionfactorsasshowninFig.10.

µ f f E

2

P

= +

L K U C

K U

Da = K Dh 2 (10)

f =

P

L

Dh

U

f

2

(9)

(11)

Using the experimental facility depicted in Fig. 5, the effective

thermal conductivities (k eff ) of the lattice structures (Lc1 – Lc4) were

measured where air was the fluid medium. The results are shown in

Fig.11.Amongthefourspecimens,Lc1hasthesmallestRhombi-Octet

unitcell of 5mmandligament width (d) of0.42mmwhereasLc4has

thelargestunitcellof12mmanddof0.99mm.Itcanbeseenthatk eff

decreaseswithincreasingunitcellsizeandincreasingdwithLc1having

thehighestk eff valueof22.8W/m 2·K.FromFig.1(a),itcanbeseenthat

all the unit cells have the same internal structure design. In order to

achieve the required unit cell size, each unit cell is scaled proportionally

in three directions. However, in order to fit the lattice structures

betweenthetopandbottomfacesheetsasshowninFig.4,theunitcell

hastobetruncated.Forinstance,theverticaldistancebetweenthetop

andbottomfacesheetsis15mm.Thisenablesthree5mmunitcellsto

bestackedverticallytoformtheLc1specimen.However,onlyoneand

ahalf10mmunitcellscanbefittedverticallybetweenthefacesheets.

Thetruncationoftheunitcellstructureshasresultedinthedifference

inporositiesofthespecimensandcouldhavecausedthedifferencesin

thek eff values.

A comparison of k eff of the Rhombi-Octet structures (Lc1 – Lc4) in

this study and the aluminum foams reported in Ref. [35] is shown in

Fig. 12, where the ligament width/diameter (d) is selected as the

characteristic length. Calmidi and Mahajan [35] presented a comprehensivestudytodeterminek

eff ofthealuminumfoamswheretheporosities

(ε) of the foams range from 0.9005 to 0.9726. This range of

aluminum foam porosities is commonly reported [36–38] and to the

best of the authors’ knowledge, there is no known reported works on

aluminumfoamswiththesameporosityastheRhombi-Octetstructure

used in forced convection heat transfer investigation. From Fig. 12, it

can be seen that for the same d value, the k eff values of the Rhombi-

Octet structures are significantly higher than those of the aluminum

foams. For instance, for d in the range of 0.55–0.6mm, the Rhombi-

Octet structure (Lc2) has a k eff value of 19W/m·K whereas the aluminum

foamhas ak eff valueof 6.7W/m·K. On the other hand,at dof

0.4–0.42mm,thek eff valueoftheRhombi-Octetstructure(Lc1)is5.5

times that of the aluminum foam. The higher k eff ofthe Rhombi-Octet

structure is likely due to its higher packing density, which resulted in

Table2

K,C E andDa 1/2 ofthevariouslatticestructures.

Specimenname L1 L2 L3 L4

K(10 −10 m 2 ) 34.0 47.6 75.5 82.2

C E 0.0446 0.0381 0.0302 0.0320

Da 1/2 0.00267 0.00316 0.00398 0.00415

282


Fig. 10.Comparison of f ⋅Da 1/2 of various Rhombi-Octet lattice structures

againstthenickelandaluminumfoams.

Fig.11.Effectivethermalconductivity(k eff )ofvariouslatticestructures.

Fig.12.Comparisonofk eff ofRhombi-Octetlatticestructuresagainstaluminum

foams.

lower porosity as compared to the aluminum foams. In addition, the

geometry of the unit cell which forms the porous matrix and their orderlyarrangementmayhavealsoresultedinmoreeffectiveconduction

pathsleadingtothehighk eff valuesofLc1–Lc4.

4.3. Forcedconvectionheattransferperformance

The forced convection heat transfer performance of the various

latticeheatsinks(L1–L4)asshowninFig.2wereinvestigatedinthe

airflowchannelofFig.7.TheexperimentalresultsareshowninFig.13

where Nusselt numbers (Nu) are plotted against the Reynolds number

(Re).AsshowninEqs.(6)and(7),NuandRewerecomputedbasedon

Fig.13.ExperimentalNuoftheRhombi-OctetlatticestructuresatdifferentRe.

the flow channel hydraulic diameter (D h ) and the experiments were

performed for Re values between 1300 and 7000. It can be seen from

Fig.13thatforeachlatticeheatsink,NuincreaseswithincreasingRe.

Inaddition,anincreaseinNuwithreducingligamentwidth(d)canalso

beobservedwherethelatticeheatsinkwiththesmallestd(L1)exhibits

thehighestNu.Forinstance,forRebetween5000and5500,L1hasa

Nuvalueof814andisapproximately38%higherthanthatofL4.Heat

transferfromtheheatedbaseisdominatedbyconductionthroughthe

latticestructurematrixandbyconvectionfromthesolidtothefluidat

theinterfaces.AsshowninFig.11,L1hasthehighestk eff ascompared

to other lattice heat sinks. This significantly improves the removal of

heatfromthebaseoftheheatsinkbyconduction.Inaddition,asshown

inTable1,theL1structureismadeofthe5mmunitcellwhichhasthe

largest surface area-to-volume ratio (A s,t /V), this also increases the

available area in contact with the fluid and enhances heat transfer by

convection.

From the above analysis, it can be observed that the average convectionheattransfercoefficient(h¯)ofthelatticestructuresareaffected

byA s,t /Vandk eff .Inaddition,inthesubsequentsection(Section4.4),it

isalsoshownthattheinterfacialheattransfercoefficient(h sf )between

the fluid and solid phases is a significant factor affecting Nu. For a

porousstructurewherethethermalconductivityofthesolidmaterialis

high,itisexpectedthatthetemperaturesoftheligamentsarecloseto

the base temperature where heat is applied [34]. Under such circumstances,

the thermal boundary layers that formed over the ligaments

and the total heat transfer area have significant influences on the

thermalperformanceoftheporousmatrix.Thetotalheattransferarea

representedbytheA s,t /Vratioisrelatedtothesizeoftheunitcelland

fromTable1,itcanbeobservedthatA s,t /Visinverselyproportionalto

d.Ontheotherhand,thecharacteristicsofthethermalboundarylayer

thatdevelopedovereachligamentdependsonthesizeoftheligament

and the distance between the adjacent ligaments (or the pore size).

Sincethefactorsaffectingtheconvectionheattransferperformancesof

the lattice structures are shown to be directly affected by d, it is

therefore appropriate to obtain a relationship between h¯ and d for all

the lattice structures investigated. On this basis, the Nusselt number

andReynoldsnumberbasedontheligamentwidth,viz.,Nu d andRe d ,

can be defined as Eqs. (14) and (15), respectively. Using the experimental

data obtained, Nu d and Re d for L1 – L4 were computed and

plotted in Fig. 14. From the figure, it can be seen that the datapoints

collapsedontoasinglelinewhichcanbedescribedbyapowerfunction.

Usingthenon-linearregressionmethod,acorrelationbetweenNu d ,Re d

andPrforthelatticestructuresisderived.Thecorrelationisshownin

Eq. (16) and plotted in Fig. 14. It can be seen that the correlation

providedagoodpredictionoftheexperimentalresultswith90%ofthe

data within±3.5% deviation from the predicted values. A maximum

deviationof9%existsatthehighestRe d valueof313.

283


c , u· T = ·[ k , · T ] + A h ( T T ) (19)

f p f f f eff f sf sf s f

0 = ·[ ks, eff · Ts ] + Asf hsf ( Tf Ts

)

(20)

kf , eff = kf

(21)

ks, eff = keff kf , eff

(22)

A uniforminlet air velocity profile, zeropressure atthe outlet and

noslipconditionboundaryconditionsatthewallswereprescribedfor

thefluiddomain.Inaddition,thefluidwasassumedtohaveauniform

inlet temperature and adiabatic conditions were imposed on the solid

and fluid domains at y=H. At y=0, where a constant heat flux was

applied, the solid and fluid phases are coupled through the boundary

conditionsofEqs.(23)and(24)[32].

Fig. 14.Plot of Nu d versus Re d of the various Rhombi-Octet lattice structures

andcomparisonagainstcorrelationofEq.(16).

= hd ¯

Nud

kf (14)

f

Re = Ud

d

µ

f (15)

Nud = 0.895Red 0.65 Pr

0.37 (16)

4.4. Interfacialheattransfercoefficient

The interfacial heat transfer coefficient (h sf ) determines the energy

transfer between the solid and fluid phases at the interface. When the

temperature difference between the solid and fluid phases cannot be

neglected, the solid-to-fluid interface significantly affects h sf which in

turninfluencestheconvectionheattransferperformancesofthelattice

structures. In order to determine h sf , a numerical simulation was performed

and the simulation results were compared against the experimental

results of the various lattice structures. The two-dimensional

computational domain shown in Fig. 15 was used for the numerical

simulation.Ithasaheightof15mmandlengthof90mmandissimilar

to the dimensions of the lattice heat sinks (L1 – L4). With the assumptionoflaminarflowandbyusingthevolumeaveragingmethod,

the steady continuity and momentum equations of the fluid phase are

written as Eqs. (17) and (18). Dukhan et al. [39] showed that, for

metallicfoams,turbulentflowisfullydevelopedforRe⋅Da 1/2 > 50.As

showninFig.10,itisreasonabletoassumethattheflowiswithinthe

laminarregimesinceRe⋅Da 1/2 ofthelatticestructuresarebetween3.5

and 30. The permeability (K) and inertia coefficient (C E ) are obtained

fromTable2andporosity(ε)of0.85whichcorrespondstotheporosity

ofaunitcellwasused.Byemployingthelocalthermalnon-equilibrium

(LTNE)model,theenergyequationsforthefluidandsolidphasescan

be derived as Eqs. (19) and (20), respectively. The LTNE model was

selected duetothe largedifference in thermalconductivityvalues between

the solid and fluid phases. In Eq. (19), k f,eff is the effective

stagnant thermal conductivity of the fluid phase and was estimated

usingEq.(21).Fromthemeasuredk eff ofthelatticestructuresshownin

Fig.11,theeffectivethermalconductivityofthesolidphase(k s,eff )can

be computed using Eq. (22). Finally, A sf denotes the interior surface

area-to-volume ratio and the A s,t/ V values in Table 1 were used. It

shouldbenotedthatinthissimulation,theeffectofthermaldispersion

wasnotconsideredasitwasfoundtobenegligiblewhenthedifference

in thermal conductivities between the solid and fluid phases is large

[35].

· u = 0 (17)

µ µ u

( u· ) u = P +

2u

K

f f f f E

2

C u u

K

(18)

q

Tf

= kf eff + k

y

, s,

eff

Ts

y

(23)

Tf = Ts (24)

The simulation was performed using the “Comsol Multiphysics”

software based on the governing equations and boundary conditions

described above. A convergence criterion of 10 −4 was prescribed and

mesh independence checks were conducted where the results were

obtained with approximately 14,000 mesh elements. For each lattice

structure, simulation was performed for the air velocities (U) ranging

from0.5m/sto5m/satintervalsof0.5m/sandwithh sf rangingfrom0

to 300W/m 2·K at intervals of 10W/m 2·K. Thereafter, the Nu values

obtainedfromthesimulationwerecomparedagainsttheexperimental

Nu values and the h sf values that produced the same numerical and

experimentalNuvaluesweredetermined.Thisprocedurewasrepeated

forthedifferentlatticestructuresandairvelocitiesandtheresultsare

plottedinFig.16,whereNu d,sf isdefinedinEq.(25).Theresultsshow

thattheNu d,sf valuesofthelatticesstructuresincreaseswithincreasing

Re d andthesevaluescollapsedontoasinglelinewhichcanbedescribed

byapowerfunction.This,therefore,suggeststhattheh sf isalsostrongly

influencedbytheligamentwidthofthelatticestructure.Byemploying

the non-linear regression method, a correlation which characterizes

Nu d,sf asshowninEq.(26)isderived.

Nu d , sf

= h sf d

k

Nu = 0.227Re Pr

d,

sf

f

0.608 0.37

d

(25)

(26)

A correlation which relates Nu d,sf and Re d in aluminum foams was

proposed by Calmidi and Mahajan [35] as shown in Eq. (27). This

correlation assumes that the fluid flow through a porous matrix is

analogous to crossflow over cylinders and employs the ligament diameter(d)ofthealuminumfoamsasthecharacteristiclengthforNu

d,sf

and Re d . In addition, as the mean fluid velocity through the porous

regionwasconsidered,the ε −0.5 termwasincludedintheequation.For

the purpose of comparison against the Nu d,sf values of the lattice

structuresobtainedfromthisinvestigation,thepropertiesofthe40PPI

aluminumfoamreportedinRef.[35]wereused,where ε=0.9132and

d=0.00025m. The Nu d,sf values were computed using Eqs. (26) and

(27)forRe d between25and300andtheresultsareshowninFig.17.

As shown in this figure, the Nu d,sf values of the aluminum foams are

consistentlyhigherthanthatoftheRhombi-Octetlatticestructures.At

Fig.15.Schematicofcomputationaldomain.

284


lattice structures have higher heat transfer performances as compared

tothealuminumfoams.Forinstance,atRe·Da 1/2 of16,thecompressed

and non-compressed aluminum foams exhibit j values of 0.125 and

0.101,respectivelywhereasthebestperformancelatticeheatsink(L1)

has a j value of 0.168. The better heat transfer performances of the

latticeheatsinksareduetheirsubstantiallyhigherk eff values(asshown

in Fig. 12) as compared to the aluminum foams. In addition, even

though the interfacial heat transfer coefficients (h sf ) of the lattice

structures are slightly lower than the aluminum foams (as shown in

Fig.17),theadvantageofhigherk eff outweighstheeffectsoflowerh sf

andresultedinhigherjvalues.

Fig.16.PlotofNu d,sf versusRe d ofthevariousRhombi-Octetlatticestructures.

low Re d , the Nu d,sf values of the aluminum foams are approximately

70% higher than those of the lattice structures. However, as Re d increases,

the difference in Nu d,sf between the lattice structures and the

aluminum foams decreases. At the highest Re d of 300, Nu d,sf of the

aluminum foams is approximately 32% higher than the lattice structures.Duetothestochasticnatureoftheporousnetwork,thealuminum

foam could have induced tortuous fluid paths which increases fluid

mixing and enhances the interfacial heat transfer rate. On the other

hand,duetotheorderlyarrangementofthelatticestructures,mixingof

the fluid may not be significant which explains the lower Nu d,sf as

compared to the aluminum foams. However, it can also be observed

fromFig.2thattheSLM-producedlatticestructureshavehighsurface

roughness.InarecentinvestigationbyHoetal.[24],itwasdetermined

thattherootmeansquare(rms)roughnessofaplainsurfacefabricated

by SLMwas approximately 7.32μm.This is significantlyrougher than

thecorrespondingvalueof0.25μmforapolishedplaincommercialAl-

6061 surface.The highlyroughsurfaces ofthelatticestructures could

have induced fluid mixing near the wall region which helped to

maintain moderately high Nu d,sf values of between 1.5 and 6.4 as

showninFig.17.

hsf

d

Nud,

sf = = 0.52 Red

Pr

k

f

4.5. Performanceevaluation

0.5 0.5 0.37

(27)

The Colburn j-factor as defined by Eq. (28) is a dimensionless

parameterofforcedconvectionheattransfer.UsingtheNu d correlation

of Eq. (16) that was derived for the lattice structures, j was computed

forthevariousairvelocitiestested.Fig.18showstheplotofjoverthe

rangeofReforlatticestructuresL1–L4.ItshouldbenotedthatReis

basedonthehydraulicdiameteroftheairflowchannel.Itcanbeseen

that j decreases with increasing Re and the rate of reduction in j also

decreases with increasing Re. In addition, j also increases with decreasing

ligament width (d) and the L1 specimen with the smallest d

exhibits the highest j values. This trend of increasing heat transfer

performance of the lattice structure with decreasing d is similarly observedinFig.13.

Fig. 19 compares the j values of the lattice structures against the j

valuesofthealuminumfoamsreportedbyBoomsmaetal.[5]andKim

et al. [34] for various values of Re⋅Da 1/2 . It should be noted that

Boomsma et al. [5] investigated on compressedaluminum foams with

waterasthecoolingmediumwhereasKimetal.[34]reportedtheheat

transfer performance of aluminum foams of 10, 20 and 40 PPI under

unidirectionalairflowanddevelopedacorrelationtocharacterizeNu.

For comparison against the lattice structures, the better performing

compressedaluminumfoamofRef.[5]wasselectedandthecorrelation

of Ref.[34]forthe40PPIaluminum foamwasconverted toanequationintermsoftheColburnj-factor.TheirresultsareshowninFig.19.

Itcanbeseenthatthejvaluesofthelatticestructuresarehigherthan

those of the aluminum foams of Refs. [5,34]. This indicates that the

j = StPr 2/3 (28)

Even though the lattice structures demonstrated significant enhancementsinheattransfercoefficients,thepressuredropsacrossthese

structures are also high. Therefore, a performance metric which accountsfortheimprovementinheattransferandtheassociatedpenalty

in pressure drop for the various structures is required. The thermal

efficiency index (η) of Eq. (29) developed by Tian et al. [19] from a

dimensionalanalysisofthepumpingpowerandheattransferisusedfor

comparisonofthedifferentenhancedsurfaces.Thisindexevaluatesthe

heat transfer performances of the enhanced surfaces at constant

pumping power and had been used for comparing stochastic metallic

foams,periodicstructures,packedbedsandlouveredfins[19].Fig.20

showstheefficiencyindicesofthevariouslatticeheatsinksfordifferent

values of Re. In addition, experiments were also performed on the cylindrical

(C1) and parabolic (P1) pin fin heat sinks of Fig. 3 to obtain

theirthermalandhydrauliccharacteristics.TheefficiencyindicesofC1

andP1weresubsequentlycomputedandplottedinFig.20.Inaddition,

using the experimental results and the respective hydraulic diameter

(D h )oftheflowchannelsreportedinRefs.[5,34],the ηandRevaluesof

thealuminumfoamswerealsocomputedbyEqs.(6),(7),(29)and(30)

andtheirvaluesarepresentedinFig.20forcomparison.Itcanbeseen

that,exceptforthealuminumfoamofRef.[34],theefficiencyindices

ofallthespecimensincreaselinearlywithincreasingRe.Theefficiency

indicesofthelatticeheatsinksaresignificantlyhigherthanthoseofthe

aluminum foams and pin fin heat sinks (C1 and P1). In addition, it is

also interesting to note that the efficiency indices of the lattice heat

sinkscollapsedontoastraightline.Thisindicatesthat,atthesameRe,

theheattransferperformanceofthelatticeheatsinksincreaseslinearly

withf 1 1/3 .

f

1/3

= Nu/ f 1

=

P Dh

L ( U /2)

1 2

f

(29)

(30)

Finally, Fig. 21 shows a comparison of the heat transfer and hydraulicperformancesoftheRhombi-OctetstructuresandtheDiamond

Fig. 17.Comparison of Nu d,sf of the Rhombi-Octet lattice structures against

metalfoamsatdifferentRe d .

285


Octet structures could likely due to the high k eff and high surface

roughnesswhichimprovesfluidmixing.

Re

Nu

H

H

f

= UH

µ

f (31)

= hH ¯

kf (32)

f

=

P H

L ( U /2)

2 2

f

(33)

5. Conclusions

Fig.18.Colburnj-factorsofRhombi-OctetlatticestructuresatdifferentRe.

Fig.19.ComparisonofColburnj-factorofRhombi-Octetlatticestructuresand

aluminumfoamsatdifferentRe⋅Da 1/2 .

Inthisinvestigation,forcedconvectionheattransferperformanceof

anewclassoflatticestructures(Rhombi-Octet)wasexperimentallyand

numerically investigated. Experiments were performed in an air flow

channelforRevaluesbetween1300and7000.Alinearheatconduction

setupwasusedtomeasuretheeffectivethermalconductivities(k eff )of

the lattice structures. Based on the local thermal non-equilibrium

model, numerical simulations were performed to understand the interfacialheattransferperformanceofthelatticestructures.Thethermohydraulicpropertiesofthelatticestructureswerecharacterizedandthe

keyfindingsofthepresentstudyaresummarizedasfollows:

1. Thepressuredrops(ΔP/L)acrossthelatticestructureswerefoundto

increase with decreasing ligament width (d). Using the

Forchheimer-extended Darcy equation, the permeability (K) and

inertia coefficient (C E ) of the lattice structures were obtained. The

permeability-basedfrictionfactors(f⋅Da 1/2 )ofthelatticestructures

areshowntobelowerthanthoseofthenickelandaluminumfoams.

The better hydraulic performances of the lattice structures are attributedtotheirlowerC

E valuesascomparedtothemetallicfoams.

2. The effective thermal conductivities (k eff ) of the lattice structures

weremeasured.Thek eff valuesofthelatticestructureswerefoundto

Fig.20.Comparisonofefficiencyindices(η)ofRhombi-Octetlatticestructures,

aluminumfoamsandpinfinheatsinksfordifferentRevalues.

and Square shaped lattice structures reported by Tian et al. [19]. It

should be noted that the hydraulic diameter in the definitions of Re H ,

Nu H and f 2 is the height of the lattice structures (H) as shown in Eq.

(31–33). Tian et al. [19] performed a systematic investigation on Diamond

and Square shaped lattices of different wire diameter and aperture

size and the lattice frame of each shape with the best thermal

performance was selected for comparison. As shown in Fig. 21 (a), at

low Re H , the friction factors (f 2 ) of the Rhombi-Octet are higher than

those of Ref. [19]. However, the friction factor of the Diamond and

SquareshapedlatticeincreaseswithincreasingRe H .ForRe H >300,the

f 2 valuesoftheDiamondshapedlatticearehigherthanalltheRhombi-

Octetstructures.Ontheotherhand,asshowninFig.21(b),Nu H ofthe

all the Rhombi-Octet structures are higher than the Diamond and

Squareshapedlattices.ThebetterthermalperformancesoftheRhombi-

Fig.21.Comparison of(a)f 2 and (b)Nu H ofthe Rhombi-Octet,Diamondand

Squareshapedlatticestructures.

286


decrease with increasing d and the highest k eff of 22.8W/m⋅K was

recorded. The k eff values of the lattice structures are significantly

higher (up to 5.5 times) than the aluminum foams. This could be

attributed to their higher packing density and orderly unit cell arrangementswhichresultedinmoreeffectiveheatconductionpaths.

3. Theinterfacialheattransfercoefficients(h sf )ofthelatticestructures

wereobtainedbycomparingthenumericalandexperimentalresults

and a relationship between Nu d,sf and Re d was established. The h sf

values areshown toincreasewith decreasingd,withL1exhibiting

thehighesth sf .However,thelatticestructuresexhibitbetween32%

and 70% poorer Nu d,sf as compared to the results for aluminum

foamsreportedintheliterature.

4. The Colburn j-factors and efficiency indices (η) of the lattice structures

were computed and compared against the results for aluminum

foams reported in the literature and conventional pin fin

heat sinks. The j values for the lattice structures are consistently

higherthanthoseofthecompressedandnon-compressedaluminum

foamsandthe ηvaluesofthelatticestructuresarealsosignificantly

higherthan theconventionalcylindricalandparabolicpinfinheat

sinks.

5. A correlation which relates Nu d and Re d of the lattice structures is

developed where 90% of the experimental results fall within±

3.5%deviationfromthepredictedvalues.Themaximumdeviation

of9%isobtainedatthehighestRe d valueof313.

6. Duetothehighforcedconvectionheattransferperformanceofthe

Rhombi-Octetlatticestructures,theycanbeusedasair-cooledheat

sinks for high heat flux electronic components such as the central

processing units (CPU) and random-access memory (RAM) chips

foundinhighperformanceserverunits.Inthisregard,thedataand

correlationpresentedinthisstudycouldbeusedaspredictivetools

bythermaldesignengineers.

Acknowledgements

The authors would like to acknowledge the assistance of Mr. Noor

MuhammadBinNoorKhalidandMs.LinLiNgforconductingsomeof

the experiments presented in this paper. The SLM equipment used in

thisresearchissupportedbytheNationalResearchFoundation,Prime

Minister's Office, Singapore under its Medium-Sized Centre funding

scheme.

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