Gas to gas heat exchanger design for high performance thermal energy storage

Gas-to-gas

heat

exchanger

design

for

high

performance

thermal

energy

storage

B.

Cárdenas*,

S.D.

Garvey,

B.

Kantharaj,

M.C.

Simpson

DepartmentofMechanical,MaterialsandManufacturingEngineering,UniversityofNottingham,UnitedKingdom

ARTICLE INFO

Articlehistory: Received5October2016

Receivedinrevisedform25January2017 Accepted1March2017

Availableonlinexxx

Keywords:

Airtoairheatexchanger Highexergyefficiency

Non-constantcrosssectionalarea Additivemanufacturing Costoptimization

ABSTRACT

Themathematicalmodellingandoptimizationofagas-to-gasheatexchangerwithanon-constantcross sectionalareaispresented.Thedesignofthecrosssectionalareaoftheheatexchangeranalyzedisbased onanhexagonalmesh,whichwouldbehighlyimpracticaltofabricateinaconventionalwaybutcouldbe builtrelativelyeasilythroughmodernmanufacturingtechniques.Thegeometricconfigurationproposed allowsattainingahighexergyefficiencyandasignificantcostreduction,measuredintermsofvolume perunitofexergytransfer.Therelationshipthatexistsbetweentheoverallexergyefficiencyoftheheat exchangeranditscostisthoroughlyexplainedthroughoutthestudy.

Theresultsobtainedfromthemodellingdemonstratethepremisethatitispossibletorealizedesigns forheatexchangersthatarehighlyexergy-efficientandverycheap,owingtothesmallvolumeofmaterial required,iftheconstrainsimposedbythelimitationsoftraditionalmanufacturingmethodsaresetaside. Furthermore, thestudyrevealsaveryimportantfact:thevolumeof materialinaheatexchanger increasesinquadraticproportiontoitscharacteristicdimension,whichimpliesthatscalingupthe geometryhasastrongimpactonitscost-effectiveness.

©2017ElsevierLtd.Allrightsreserved.

1.Introduction

Heatexchangers(HX)areextensivelyusedindiverseindustrial processesnowadays. Several differenttypes of heatexchangers havebeendeveloped fordifferentapplications;beingthe shell-and-tubeandplate-finthemostcommonlyutilizedconfigurations [1].

ThedesignofaHXinvolvesanumberofhighlyinterdependent geometricandoperatingvariablesthatoftenexhibittrade-offs[2] however,throughacarefulselectionofparameterscost-effective designs witha high efficiencycan be realized.This is of great importancefortheindustryandinparticular,forthedevelopment and widespread deployment of utility-scale energy storage technologies, such as compressed air energy storage (CAES) systems.

Avastamountofresearchhasbeendedicatedinpastyearsto developdesignstrategiesforachievingsubstantialreductionsin thecostof aHX.The useof evolutionaryalgorithmsand other population-basedoptimizationmethodsforminimizingthetotal costofHXs hasbeenstudiedbymany researchers.Following a

literature reviewcan befound,inwhich examples ofthework aimedat optimizingthedesign of HXsin termsof maximizing performance and/or minimizing cost carried out by different authorsarediscussed.

Sanaye and Hajabdollahi carried out by meansof a genetic algorithmtheoptimizationintermsofcostandeffectivenessofa shell-and-tube HX[3]and aplate-finHX[4]presentingin both studiesasetofmultipleoptimumsolutions(Paretofront)forthe objectivefunctions.Inadifferentstudy,Hajabdollahietal.[5]used ageneticalgorithmtooptimizeacompactplate-finHXintermsof effectivenessandpressuredrops.Theauthorsobservedthatany geometric changethat reducespressure drops in the optimum situationhasanegativeimpactintheeffectivenessoftheHXand vice versa, therefore a set of Pareto optimal solutions was presented.Similarly,Najafietal.[6]carriedouttheoptimization of aHXofthesametype, albeitfocusingondifferentobjective functions. Theresearchers providea number of Paretooptimal solutions, which exhibit the trade-off that exists between the maximumtotalrateofheattransferandtheminimumtotalcostof thecomponent.

Patel and Rao used a particle swarm optimization (PSO) algorithm tominimizethetotal annualcostofashell-and-tube [7] and a plate-fin HX [8]. Several different case studies are presented throughwhich the effectivenessand accuracyof the * Correspondingauthor.

E-mailaddress:Bruno.Cardenas@nottingham.ac.uk(B. Cárdenas).

http://dx.doi.org/10.1016/j.est.2017.03.004

2352-152X/©2017ElsevierLtd.Allrightsreserved.

–

ContentslistsavailableatScienceDirect

Journal

of

Energy

Storage

algorithmisdemonstrated.Theresultsshowanimprovementwith respecttotheresultsobtainedbypreviousresearchersviamore standardgeneticalgorithms.Furthermore,Sadeghzadehetal.[9] presentedacomparisonbetweenthePSOandageneticalgorithm for the optimization of a shell-and-tube HX in terms of cost, expressedasafunctionofsurfaceareaandpowerconsumption. ThePSOmethodproducedsuperiorresultsfortheproblem,which supportstheconclusionsfromPatelandRao[7,8].

Someauthors have explored theuse of variantsof thePSO method.Marianietal.[10]optimizedthedesignofa shell-and-tube HX through a quantum PSO algorithm. The researchers reported that considerable reductions in capital investment (20%)andannualpumpingcost(72%)wereachieved.Turgut [11],ontheotherhand,investigatedtheuseofahybrid-chaotic PSOalgorithmforthemulti-objectiveoptimizationofaplate-fin HXintermsofheattransferarea,totalpressuredropsandtotal cost.TheauthorobservedthatthePSOalgorithmproducedmore accurate results than many other optimization algorithms discussedintheliterature.

Besidesthetechniques aforementioned, several other evolu-tionaryandpopulationbasedoptimizationalgorithmshavebeen usedbymanyresearchersforoptimizingthedesignofHXsfrom differentperspectivessuchastheimperialist-competitive, cuckoo-search,biogeography-based optimization,teaching-learningand fireflyalgorithms,amongothers.

Hadidietal.[12]optimizedthedesignofashell-and-tubeHXby means of an imperialist-competitive algorithm. The authors achievedreductionsincapitalcostofupto6.1%withrespectto selected reference cases. Yousefi et al. [13] used an improved harmony-searchalgorithmtooptimizethedesignofaplate-finHX withaimsatminimizingheattransferareaandpressuredrops.The resultsobtainedindicatethattheapproachstudiedcanproduce moreaccuratesolutionsthangeneticandPSOalgorithms.

Wang andLi [14]carried out amulti-objective optimization (maximizingefficiencyandminimizingcost)ofaplate-finHXviaa cuckoo-searchalgorithm(CSA).Theauthorsreportedsatisfactory resultsandconcludedthattheCSAmethodiscapableofgenerating moreaccurate solutionsthan single-objectiveapproacheswhile requiringlessiterations.Likewise,Asadietal.[15]usedaCSAfor theminimizationofthecostofashell-and-tubeHX.Theauthors achieved reductions in cost of 9.4% and 13.1% with respect to resultsproducedbygeneticandPSOalgorithms,respectively.

HadidiandNazari[2]carriedoutacostoptimizationofaHX throughabiogeography-basedoptimization(BBO)algorithmwith which reductions in capital investment and operating costs in comparison toliterature referencecases of upto14% and 96%, respectively,wereachieved.Pateland Savsani[16]andRaoand Patel [17] carried out, by means of a modified version of the teaching-learning algorithm, a multi-objective optimization fo-cusedoneffectiveness and totalcost ofa shell-and-tube and a plate-finHX,respectively.TheauthorsreportedthatbetterPareto optimal solutions were attained than those found by generic algorithmsforananalogousproblem.

Morerecently,Mohanty[18] publishedtheresultsofthecost optimization of a shell-and-tube HX carried out via a firefly algorithm.Thestudyshowsthatthetotalsurfaceareaandtotal costcanbereducedby27%and29%,respectively,withrespectto thereferencedesigns.Furthermore,thestudypresentsa compari-sonofthefireflyalgorithmagainstsomeof theaforementioned

Nomenclature

Acronyms

HP highpressure HX heatexchanger LP lowpressure

Symbology

A crosssectionalarea(m2₎

As heattransferarea(m2₎

B meanexergytransfer(W/m) _

BHP exergyoftheheatattheHPside(W/m)

_

BLP exergyoftheheatattheLPside(W/m)

_

BLOSS totalexergylosses(W/m)

_

BLQ exergylossduetoheattransfer(W/m)

_

B_DP exergylossduetopressuredrop(W/m)

_

B_DHP exergylossduetopressuredropintheHPside(W/m)

_

B_DLP exergylossduetopressuredropintheLPside(W/m)

CP specificheatcapacity(J/kgK)

D

P pressuredropperunitlength(Pa/m)

D

PHP pressuredropintheHPside(Pa/m)

D

PLP pressuredropintheLPside(Pa/m)

D

T temperaturedeltafromTavg(K)

D distancebetweencentresofHPpipes(m)

e

roughnessheight(m)

e

/; relativepiperoughness fD Darcy–Weisbachfrictionfactor h convectioncoefficient(W/m2_K)

kair thermalconductivityofair(W/mK) kwall thermalconductivityofwall(W/mk)

l

fractionofHPpipeperimetercoveredbyflanges L heightoftheflange(m)

m

dynamicviscosityofair(Pas) _

mHP massflowrateperHPpipe(kg/s)

_

mLP massflowrateperLPpipe(kg/s)

n numberofringsofHPpipes Nu Nusseltnumber

; pipediameter(m)

c

fractionofthepipebeinganalyzed p perimeterofflowareaofpipe(m) P pressure(Pa)

PHP pressureoftheHPside(Pa) PLP pressureoftheLPside(Pa) Pr Prandtlnumber

_

Q heattransferrateperunitlength(W/m)

r

densityofair(kg/m3₎

rHP radiusoftheHPpipe(m) Re Reynoldsnumber

S allowablestressofpipematerial(Pa) tHP thicknessofHPpipes(m)

tLP thicknessoftheflangeatthebase(m) tmf thicknessoftheflangeatmidpoint(m) Tamb ambienttemperature(K)

Tavg averagetemperatureofHXsection(K) THP temperatureoftheHPstream(K) Ti temperatureofinnerwallofHPpipe(K) TLP temperatureoftheLPstream(K) Tmf temperatureoftheflangeatmiddle(m) TO temperatureofouterwallofHPpipe(K) TðxÞ temperatureatapointxinflange(K) rT temperaturegradientofHXsegment(K/m) U meanflowvelocityofair(m/s)

V=B volumeofmaterialperunitexergytransfer(m3_/kW)

W exergyefficiency

X ratioofproportionalpressuredrops Y ratiooftemperaturedifferences

Z fraction of total exergy losses caused by pressure drops

optimizationtechniques suchas GA,PSO, CSAand BBO, which suggeststhatthefireflyalgorithmisthemosteffectivemethodfor thecost-optimizationofthedesignofaHX.

Itisclearthatasubstantialamountofresearchhasbeencarried outwiththeendgoalofreducingthetotalcostofHXs;numerous researchershavestudiedseveraldifferentoptimizationalgorithms throughtheuseofwhichverygoodresultscanbeachieved.Allthe optimizationtechniques abovementioned canachieve consider-able reductions in cost despite the fact that their outputs are restrictedbyapre-conceiveddesign;however,betterresultscould be achieved by the optimization process if the starting point (design)wasbetter.

Theheatexchangers thatarecurrentlyusedbytheindustry weredesigneddecadesagounderthelimitationsofthefabrication methodsofthosetimes.Nevertheless,thereisamuchwiderrange ofmanufacturingtechniquesavailablenowadays,suchasadditive manufacturing,thatarecapableofproducingmuchmorecomplex designs,throughwhichhigherefficienciesand/orlowercostscan be achieved. Hence, research efforts should be focused on developinginnovativedesignsofHXs,notbasedonexistingones, thatexploitthecapabilitiesofmodernmanufacturing.

Accordingly,thisworkpropoundsandinvestigatestheuseofan intricategeometry,whichwouldbehighlyunpracticaltofabricate bytraditionalmethods,foragastogasheatexchangerwithaimsat obtaining designs that simultaneously maximize the exergy efficiencyand minimizethecost. Themain differencebetween thepresentstudyandtheinvestigationsdiscussedbeforeisthat ratherthan simplytryingtotune geometricand/oroperational parametersofanexistingdesignofHXtoachieveareductioninthe costoftheequipment,a ratherradialdesign isproposedasthe startingpoint.Itisimportanttohighlightthatthisworkfocuses particularlyonunderstandingthetrade-offthatexistsbetweenthe overallexergyefficiencyoftheHXanditscost.

2.Geometricdesignofthegas-to-gasHX

Generally,HXshaveaconstantcross-sectionalareathroughout theirwholelength;however,itwouldbemuchbettertohaveaHX withgradedproperties.Theapproachfollowedinthepresentwork forfindingtheoptimumgeometry,intermsofmaximumefficiency andminimumcost,istodesignaHXwithanon-constant cross-sectionalareabetweenthehotandcoldends.Forthispurpose,the lengthoftheHXisdividedintondifferentsegments(orslices)and eachsegmentisdesignedspecificallyforthetemperaturerangeat whichitwilloperate.

Thebodyoftheheatexchangercanberegardedasseriesof segmentswhosedimensionsadjustwithlength.Eachsegmenthas aconstantcross-sectionthroughoutitslength(whichshouldbe relativelysmall)however,geometricparameterschangefromone segmenttothenext.Therefore,thepropertiesoftheHXasawhole aremodified indiscrete stepsratherthan varyingcontinuously withlength;although,thereisnoreasonwhythelattercouldnot bethecase.

Thecross-sectionalareaofeachofthesegmentsisbasedonan hexagonalmesh(honeycombpattern).Thisgeometrywaschosen becauseithasmultipleaxesofsymmetry,whichallowsanalysinga small section of it assuming that a similar behaviour can be observedelsewhere.

The low pressure (LP) and high pressure (HP) pipes are distributedsothat each HP pipeis surrounded bysix LP pipes whereaseachLPpipeissurroundedbythreeHPpipes,asshownby Fig.1.Thecross-sectionalareaofthesegmentofHXstartsasafully hexagonalmesh.TheregionscorrespondingtotheHPpipesare transformedintocircles,becauseacylinderisthebestgeometric shapeforapipethatwillhandlepressurizedfluids,theouterwall

ofthesepipesisalsorounded(althoughitcouldremainstraight)to makeanefficientuseofmaterial.

The arrowsin Fig.1 show the heat flows assumed for the modelling.EachHPpipedeliversheattothesixadjacentLPpipes whereaseachLPpipereceivesheatfromthethreeHPpipesaround it.ItisassumedthatnointeractionoccursbetweencontiguousLP pipes.Anadvantageouspropertyofthegeometryuponwhichthe cross-sectionaldesignisbasedisthatitgrowsinconcentricrings, asshownbyFig.2.ThischaracteristicallowsaddingringsofHP pipesattheperimeterofthecrosssectiontoincreasethecapacity oftheHXwithoutmodifyingitsbehaviour.ThenumberofHPand LPpipesinthearrangement,accordingtothenumberofrings(n), isgivenbyEquations(1)and(2),respectively.

HP¼1þX

n

i¼1

6i ð1Þ

LP¼X

n

i¼1

12i6 ð2Þ

HP

LP

n=1

n=2

Fig.2. Growthofthecross-sectionoftheHXinconcentricrings.

AsectionofthegeometryconsistingofthreeHPpipesandthe LPpipecomprisedbetweenthemisshowninFig.3.Asitcanbe seen,thewallsoftheLPpipearemadebytrapezoidalelements, whichwillbereferredtoas“flanges”henceforth.Itisassumedthat theLPstreamofthegastogasHXwillbeatambient(ormarginally higher)pressure; therefore,theflangesdo nothaveathickness requirementtowithstandpressure.Thethinnestpossiblewallis desirable from a material utilization standpoint; however, the flangesneedtobethickenoughtoholdthestructuretogetherand totransportheatfromthewalloftheHPpipetotheLPstream. Stateddifferently, theirsizing isconductiondrivenratherbeing dictatedbypressure.

Therearefivegeometricparametersthatdefinethegeometry, ascanbeseeninFig.3:ThedistancebetweencentresofHPpipes

D

ð Þ, the radius ðrHPÞ and thickness ðtHPÞ of the HP pipes, the

thicknessoftheflangeatthebaseðtLPÞandthethicknessofthe

flangeatmiddleheight tmf

.Forsimplicity,thethicknessofthe flangeatmiddleheightissomewhatarbitrarilydefinedtobeafifth ofthethicknessatthebase.

3.MathematicalmodellingofanHXsegment

Theobjectiveis,asaforementioned,tofindanoptimumdesign intermsofcostperunitofexergytransferforthecross-sectional areaofaspecificsegmentofaHXforthetemperaturerangeat whichitwilloperate.Oneofthemaincostdrivingfactorsinan additivemanufacturingprocessisthecostofthematerialutilized; henceachievingreductionsintheeffectivevolumeofthedesignis a good approach towards minimizing the total cost of the equipment. The mathematical model and algorithm developed arefocusedonthedesignandoptimizationofonly1segmentof theHX.TheoptimizationofthewholebodyofaHXcanbedoneby applyingthesamealgorithmtoeachoneofthensegmentsinthe structureandoptimizingthemfortheircorrespondingoperating temperature.

Anexhaustivesearchbasedontestingeverypossible combina-tionofvariablestofindtheoptimumdesignisunquestionablynot agoodapproachbecauseagreatdealofallpossiblecombinations ofvariableswillfailtomeettherequiredperformanceormaybe physically unrealizable, in addition of being computationally expensive. Therefore, an organized approach to explore the multidimensionalspacecontainingthesolutionisrequired.

Althoughseeminglycounterintuitive,fouradditionalvariables, named W, X, Y and Z are introduced. These variables help to constrainthe multidimensional spaceby relating two or more

operationalparametersoftheHXsegmentbetweenthem.Thenew variablesaredefinedinthefollowingway:

Wisthedesired exergyefficiencyof theHXsegment. Inthis studyefficiencies in therange of 97–99% are evaluated.This parameterdefinesindirectlytheaveragetemperaturesoftheHP andLPstreams,asitwillbeexplainedfurtheron.

X,givenbyEq.(3),istheratioofproportionalpressuredrops(

D

P) betweenthetwoairstreams.

Y,givenbyEq. (4),describesthetemperaturedropalong the flangesofthestructure.

Z,givenbyEq.(5),istheratiooftheexergylossesduetopressure dropswithrespecttothetotalexergylosses.

X_¼

D

PHP PHP

=

D

PLP

PLP

ð3_Þ

Y¼TmfTLp

ToTLP ð

4Þ

Z_¼BDHPþBDLP

BLOSS ð

5_Þ

Thefourpreviouslydescribedfactorsarecalled“slowmoving variables”becausetheirvaluechangeslittleincomparisonwith therestofthe(fastmoving)variablesinthespacesuchasD,rHP, tLPandm;_ thusitispossibletodefinetheirvaluearbitrarilyand stillapproximateagoodsolution.Thevaluesoftheslowmoving variables can be subsequently fine-tuned to achieve a further reductioninthevolumeofmaterialrequiredperunitofexergy transfer.

Achallengeencounteredwithinthepresentstudyisthatallthe variablesarehighlyinterdependentbetweenthem;wherebythe implementationofaniterativeprocesstosearchforthesolutionis necessary. Thealgorithmdevelopedfor thispurposeconsistsof three iterative loops, one that determines the radius and wall thicknessoftheHPpipessothattheproportionalpressuredrops satisfyEq.(3),asecondonethatdeterminesthethicknessofthe flangesbasedonEq.(4)andathirdouteronethatencompassesthe previoustwoandadjuststhemassflowratessothatEq.(5)ismet. Theprocessfollowedbythealgorithmisgraphicallyexplainedin Fig.4.

ThedistancebetweencentresofHPpipes(D)isthefastmoving variable that is used as the characteristic dimension.Different designsare obtainedfora number ofdifferentvalues ofDand comparedbetweenthemtofindthebestperformingconfi gura-tion. An initial set of parameters has to be provided to the algorithm before any calculations are performed. This set comprises:

The average temperature at which the HX segment will be operatingTavg

The pressures of the HP and LP streams (PHP and PLP, respectively)

Thermal conductivity ðkwallÞ and allowable stress ðSÞ of the

materialoftheconstruction

After theinitial parameters, the values of theslow moving variablesW,X,YandZaredefined,aswellastheDforwhicha designwillbeobtained.ThetemperaturesoftheHPandLPfluids aregivenbyW,ZandTavgthroughthefollowingsetofequations:

_

BLP¼Q_ 1

Tamb

Tavg

D

T !

ð6Þ Fig.3.Sectionconsistingof3HPpipesandtheLPcomprisedbetweenthem.

_

BHp¼Q_ 1

Tamb

Tavgþ

D

T !

ð7_Þ

B¼B_LPþB_Hp

2 ð8Þ

1W

ð Þð1ZÞ¼B_HPB_LPB ð9Þ

THP¼Tavgþ

D

T ð10Þ

TLP¼Tavg

D

T ð11Þ

The_firststepofthecalculationprocessistotakeaguessforthe mass flow rates. The HX has equal mass flow rates in both directions,whichisveryconvenientfromanengineeringpointof view.Therefore,themassflowrateinaHPpipeðm_HPÞisdouble

than the mass flow rate in a LP pipe ðm_LPÞ becausethere are

twiceas many LP pipes than HP pipes. It is also important to highlight that performance parameters in the calculations are expressedperunitlength,asthelengthofthesegmentofHXisleft undefined.

The radius of the HP pipe ðrHPÞ is determined by the first

iterativeloopofthealgorithmsothattheproportionalpressure dropsofbothstreamssatisfyEq.(3)fortheXdefined.Forclarity,a valueof1forXmeansthattheproportionalpressuredropsofboth streams are equivalent while ifX >1the proportional pressure lossesintheHPstreamaregreaterthanintheLPstream.First,a guessforrHPismade;thenthepressuredrops(

D

P)perunitlength inside thepipesarecalculatedbasedonthis valuebymeansof Equation(12) [19].Thefrictionfactor ðfDÞis calculatedthrough

Eq. (13), which is an approximation to theiterative Colebrook equation[20].Theaccuracyofthefrictionfactorobtainedfromthis equationhasanerrormarginof2%forReynoldsnumbersgreater than3000[21].

D

P¼fD

r

U

2 ð2

_f

Þ1

ð12Þ

fD¼ 1:8log10 ð

e

=3:7;Þ 1:1

þ6:9Re1

h i

2

ð13Þ

ThecaseoftheHPstreamiscalculatedfirst.AtthispointtHPis alsodetermined,bymeansofEq.(14).Thethicknessofthepipe dependsonthepressureofthefluidandontheallowablestress(S) of thematerial[22]. Itshouldbementioned thattheallowable stressisafixedvaluedefinedatthebeginningofthecalculations, thereforetheoptimizationalgorithmcannotgeneratedesignsthat do nothaveawallthickenoughtoholdthepressureoftheHP stream.Next,thepressuredropontheLPsideiscalculated.Given Fig.4.Algorithmusedforthecalculations.

that the cross sectional area of the LP pipe is not circular, an equivalent hydraulic diameter has to be calculated through Eq.(15).

tHP¼rHPPHP

SPHP ð

14Þ

f

h¼4A=p ð15Þ

ThevalueofrHPisupdatedandthecalculationsarerepeated untilthevaluesobtainedfortheproportionalpressuredropson bothsidessatisfyEq.(3)forthevalueofXdefined.Subsequently, theseconditerativeloopofthealgorithmcalculatestherequired thicknessfortheflanges.Thisisdonethroughadoubleiteration withTO,whichisthetemperatureoftheouterwalloftheHPpipe, andtLP,whichisthethicknessofthebaseoftheflange.Guessesare madeforbothvariables anda Tmf isfoundsuchthat Eq.(4) is satisfiedforthevalueofYdefined.Theflangeisassumedtohavean internaltemperaturegradientdescribedbyEq.(16)[23].

TðxÞ¼YðTOTLPÞcosh cosh

1 1

Y

L_Lx

þTLP ð16Þ

Asaforementioned,theairineachLPpipeisheatedbytheair flowingthroughthreedifferentHPpipes.Duetothesymmetryof thehexagonalmeshitispossibletoassumethat1/6oftheairina HPpipeinteractswith1/3oftheairinaLPpipeandthatthemass of LP air contained in that 1/3 of pipe does not interact with remaining2/3ofLPair.Withbasisonthisassumption,itispossible todividethegeometryevenfurther,asshowninFig.5,sothatonly a1/12ofaHPpipeanda1/6ofaLPpipeareanalyzed.

It isimportanttokeepinmindthatthebehaviouroftheHP pipesoftheouterringofthecross-sectionoftheHXisslightly differentfromwhatisobservedintherestofthepipesoftheHX. ThisisduetothefactthatinthecaseoftheHPpipesoftheouter ring only 2/6 of their surface is surrounded by LP pipes (the remaining 4/6 would be probably insulated in a practical application) whereas the rest of the HP pipes of the HX are completelysurroundedbyLPpipes.Consequently,theapproachof onlyfocusingonaverysmallsectionofthecross-sectionalareaof theHXforthecalculations(asshowninFig.5)isonlyvalidifthe HXis madeup bya largenumber of rings(n>12) so thatthe differenceinconditionsattheouterringdoesnothaveasignificant effectontheresultsobtained.

Theheatflows(perunitlengthofHX)insidethisfractionofthe geometry are calculated by means of the one dimensional conduction equation (Eq. (17)) and the convection equation

(Eq.(18))asappropriate.

_

Q¼kA

d

T

d

x ð17Þ

_

Q_¼hAs

d

T ð18Þ

Forexample,Q3_ isaheatflowbyconductionfromthewallof theHP pipetotheflange,which is determinedbythethermal conductivity of the material of the construction, the heightof flangeuptoitsmiddlepoint(L),thecross-sectionalareaofthe flange(importanttonotethatdue totheshape,thisareaisnot constant along the height of the flange), the temperature difference between the outer surface of the HP pipe ðToÞ and

thetemperatureoftheflangeatmiddleheight Tmf

.Ontheother hand,Q_2isaheatflowbyconvectionfromthewalloftheHPpipe totheLPfluid,whichisdeterminedbythetemperaturedifference between the outer surface of the HP pipe and the mean temperatureoftheLPfluid,thecontactareabetweentheHPpipe wallandtheLPfluidandaheattransfercoefficientðhÞ.

TheheattransfercoefficientsarecalculatedthroughEq.(19).It isworthmentioningagainthatforthecaseoftheLPstreaman equivalenthydraulicdiametershouldbeusedasthepipeis not cylindrical.

h¼Nukair

f

ð19Þ

where

Nu¼0:023Re0:8_pr0:33 _ð20Þ

Re¼m

f

m

A ð21Þ

Pr¼

m

Cp

kair ð

22Þ

ThespecificheatcapacityoftheairðCPÞiscalculatedthrough

theequationproposedbyLemmonetal.[24]whilethedynamic viscosityð

_m

ÞandthermalconductivityðkairÞarecalculatedthrough

theLemmonandJacobsenequations[25].

Ifanenergybalanceinthissectionisnotmet,thevaluesfortLP andTOarerevisedandcalculationsarerepeated.Atthispoint,the geometryhasbeencompletelycharacterized andtheheatflows areknown,thusitispossibletocalculatethetemperaturegradient oftheHXsegmentthroughEq.(23),where

_c

isthefractionofthe pipebeinganalyzed(1/12forthecaseofHPpipesand1/6forthe caseofLPpipes).Theexergylossesduetopressuredropscanbe calculatedthroughEq.(24),theexergylossesduetoheattransfer bymeansofEq.(25),thetotalexergylosses(26)whilethemean exergytransferisgivenbyEq.(8).

rT¼_{m_} Q_

ð

c

ÞCp ð

23Þ

Table1

Setofinitialparametersforthecasestudy.

Slowmovingvariables Propertiesofair Propertiesofsteel

W 0.97,0.98&0.99 Tavg 745K&400K Kwall SeeEq.(27)(W/mK)

X 1.0 Tamb 290K S 110.6MPa

Y 0.5 PHP 5MPa e/; 0.005

Z 0.5 PLP 101.325kPa

Fig.5.Sectionofthegeometryusedforanalysiscomprisedby1/12oftheHPpipe and1/6oftheLPpipe.

_

B_DP¼ð

c

Þm_ RTamb

D

P=P

ð24_Þ

_

BLQ¼B_HPB_LP ð25Þ

_

BLOSS¼B_LQþB__DHPþB_DLP ð26Þ

IfEq.(5),whichdictatestherelationshipbetweenexergyloses duetopressuredropsinbothstreamswithrespecttototalexergy lossesisnotsatisfied,thevaluesform_HPandm_LParerevisedand

thewholeprocessiscarriedoutagain.AfterknowingthevalueofB itispossibletocalculatethevolumeofmaterialrequiredperunit of exergy transfer, which is the objective function of the optimization.

4.Analysisofresultsobtainedanddiscussion

A case study carried out through the algorithm previously described is presented and thoroughly analyzedin thecurrent section.ThematerialselectedforthestructureoftheHXisatypical stainlesssteel,whosethermal conductivityis givenbyEq.(27). Materialswithahigherthermalconductivityorlowercostcould beused.

Kwall¼25:5þ

Tavg

80 T2avg

120103 ð27Þ

20differentdesignscorrespondingto20differentvaluesofD, going from 0.01m to 0.25m, were generated. This allows understanding the behaviour of different performance metrics oftheHXasitisscaledupandchoosingtheconfigurationthat attainsthehighestexergytransferperunitvolume.Twodifferent averagetemperaturesareanalyzed:745Kand400K.Oneofthe envisagedapplicationsfor aHXof thiskindarecompressedair energystoragesystemsinwhicha highexergyefficiencyof the componentsisofparamountimportance.Typicallythesesystems producestreamsofcompressedairatpressuresintherangeof5–

7MPaandtemperatureintherangeof770–800K.AsegmentofHX operatingatanaveragetemperatureof745Kwouldbenearthehot endofthedeviceinsuchanapplication.Thetemperatureof400K selectedfor second case study was selected (ratherarbitrarily) with the aim of investigating how the different performance parametersvarywithtemperature.Theparametersandvaluesof theslowmovingvariables usedfor thecase studyare givenin Table1.

Fig.6showsthevariationofthevolumeofmaterialrequiredper unitofexergytransferwithrespecttoDatdifferentlevelsofexergy efficiency.Twoimportantfactscanbeobserved.Thefirstisthat morematerialisneededperkWofexergytransferasDincreases; thus the performance of the HX worsens. For Tavg¼745K,

D=0.01m and W=0.99 a volume of 17.16105m3 per kWof exergytransferisneeded,whileforaD=0.25mthevolumeneeded atthesameexergyefficiencyincreasesto70.41103_m3_.

IftheresultspresentedinFig.6areplottedinalog-loggraphit isfoundthatthecurveshaveanaverageslopeof1.949,1.922and 1.869forW=0.97,0.98and0.99respectively,atatemperatureof 745K;whileforatemperatureof400Ktheslopesare1.69,1.769 and1.811.TheforegoingmeansthatthevolumeperkWofexergy transfer nearly quadruples if the characteristic dimensionD is doubled, which is a very important observation. This is in accordance with empirical conclusions from the HX industry whichsuggestthatingeneralsmallerdimensionsyieldabetter performance.

ThesecondimportantfactthatcanbeobservedinFig.6isthat morevolumeofmaterialisrequiredperkWofexergytransferas

the exergy efficiency level increases; i.e. a HX becomes (as expected)moreexpensiveasitbecomesmoreefficient. Addition-ally,itmaybeseenthatthedifferenceinvolumefromonevalueof WtoanotherincreasesasWgetsbigger;however,theincreaseis notaslargeasexpected.

Fig.7showsthevariationofthemassflowratesfordifferent exergyefficienciesasthegeometryisscaledup.Onlym_HPvalues

areshownasithasbeenestablishedthatm_LPishalfasmuch,to

haveequivalentmassflowratesinbothdirections.Themassflow ratesperpipeincreasewithD,asexpected,becausethegeometry becomesbigger.ForTavg¼745K,D=0.01mandW=0.99am_HPof

24.26105_{kg/sisneededwhileforaD=0.25mandthesameW}

them_HPneededincreasesto11.79102kg/s.

Aninterestingfactisthatthemassflowratesincreaseasthe exergyefficiencylevel(W)decreases.Thisoccursbecauseasthe geometrygrowsthe

D

Pinthepipesdecrease;

D

Palsodecreasesas Wincreases.Theforegoingmeansthathigherm_ arerequiredto satisfyEq.(5);inotherwords,thevalueofZforcesthesystemto increasem_ sothatmoreexergyislostduetopressuredropsandthe proportionestablishedbyEq.(5)canbemet.

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Volume of Material per kW of Exergy Transfer (m3/kW)

10-5 10-4 10-3 10-2 10-1 100

T=400K W=99% T=400K W=98% T=400K W=97% T=745K W=99% T=745K W=98% T=745K W=97%

Fig.6.VolumeofmaterialperkWofexergytransferfordifferentvaluesofD.

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Mass Flow Rate per HP pipe (kg/s)

10-4 10-3 10-2 10-1

T=400 W=99% T=400 W=98% T=400 W=97% T=745 W=99% T=745 W=98% T=745 W=97%

Fig.7.MassflowrateperHPpipefordifferentvaluesofD.

ThebehaviourofthevelocitiesofbothstreamswithrespecttoD fordifferentlevelsofexergyef_ficiencyataTavgof745and400Kis

shownin Figs. 8 and 9,respectively. Velocitiesdecrease asthe averagetemperatureofthesegmentofHXdecreases.Inbothcases, velocitiesdecreaseasDincreases(despiteincreasingmassflow rates)duetobiggerpipediameters.Thevelocitiesofthestreams decreasewith increasing W because pressure drops (shown in Figs.10and11)dependdirectlyonthem,sovelocitieshavetobe reducedinordertokeeppressure-relatedexergylosses B__DP

toa

minimumandachievehigherexergyefficiencies.

Thevariationofthetemperaturegradient(temperaturechange ofthestreamsperunitlength)ofthesegmentofHXwithrespectto DatdifferentlevelsofexergyefficiencyisshowninFig.12.This parameter is of great importance as it provides information regarding the length of the HX. The temperature gradient decreases with increasing D because m_ increases. For

Tavg¼745K, D=0.25m and W=0.99 a rT of 0.5354K/m is

observedwhileaconsiderablyhigherrTof25.25K/misachieved forD=0.01mandW=0.99.

Lowerefficienciesexhibitahighertemperaturegradientdueto alargerheattransferresultantfromincreasedmassflowratesand heat transfercoefficients. Thisisin agreement withtheresults presentedinFig.5,whichstatesthatmorevolumeofmaterialis required per kW of exergy transfer as the exergy efficiency increases.

The fraction of theouter perimeter of theHP pipes thatis covered by flanges ð

_l

Þ increaseswith D,which contributes to the larger volume per kW of exergy transfer observed for largervaluesofD.ForaTavg¼745K,D=0.01mandW=0.99a

l

of0.027isobservedwhile

_l

increasesto0.285forD=0.025mand W=0.99.ItshouldalsobenotedthatforanygivenvalueofD,the HPpipesareless coveredby_flangesastheef_ficiencyof theHX increases.

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Pressure Drops in the HP and LP sides (Pa/m)

10-1 100 101 102 103 104

T=745 W=97% HP T=745 W=97% LP T=745 W=98% HP T=745 W=98% LP T=745 W=99% HP T=745 W=99% LP

Fig.10.Pressuredropsinbothstreams(HPandLP)oftheHXfordifferentvaluesof D,T=745K.

Fig.11.Pressuredropsinbothstreams(HPandLP)oftheHXfordifferentvaluesof D,T=400K.

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Velocity of air in the LP and HP sides (m/s)

2 4 6 8 10 12 14 16 18

T=745 W=97% HP T=745 W=97% LP T=745 W=98% HP T=745 W=98% LP T=745 W=99% HP T=745 W=99% LP

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Velocity of air in the LP and HP sides (m/s)

0 0.5 1 1.5 2 2.5 3

T=400 W=97% HP T=400 W=97% LP T=400 W=98% HP T=400 W=98% LP T=400 W=99% HP T=400 W=99% LP

The aspect ratio (Area of HP pipe/Area of LP pipe) of the segmentofHXincreaseswithD,meaningthattheHPpipesgrow fasterincomparisontoLPpipes;however,theincreaserateofthis ratiodecreasesasDbecomeslarger.ForaTavg¼745K,D=0.01m

and W=0.99 an aspect ratioof 0.0686 is observed while for a D=0.25mandW=0.99thisvalueincreasesto0.0719.Foranygiven Dthisratioissmallerforthehigherefficiencydesignsbecausefor achievinghigherexergyefficienciesexergylossesduetopressure dropsareminimizedbyreducingmassflowrates,whichtranslates intosmallerradiioftheHPpipesandconsequentlyasmalleraspect ratio.

Oneofthemainbenefitsofferedbytheslowmovingvariablesis theabilitytocarryoutanoptimizationprocessthroughtheirfine tuningto improvethe resultsobtainedfrom themathematical model.Theoptimizationprocessemployed,graphicallyshownby Fig.13,isknownastheone-factor-at-a-timemethod.Itoperatesin thefollowingway:

ThedesignforanyDcanbechosenastheinitialpoint.Asweep throughdifferentvaluesofXisdone;maintainingDandtherest

oftheslowmovingvariablesfixedtoidentifythevalueofXthat yieldsthebestperformance.

Thereafter, a sweep through different values of Y is done; maintainingfixedtheD,W,ZandthenewfoundvalueofX,to identifythevalueofYthatyieldsthebestperformance. Subsequently, a sweepthrough differentvalues of Zis done;

maintainingfixedtheD,WandthenewfoundvaluesofXandY,to identifythevalueofZthatyieldsthebestperformance. Thiscompletesthefirstiteration;theprocessisrepeateduntil

thevaluesforX,YandZdonotchange.

ThedesignforD=0.01m,W=0.99andTavg¼745Kisusedas

the starting point for the one-factor-at-a-time optimization. A value1.716104_m3_{/kWwasobtainedwiththeoriginalvaluesof}

theslowmovingvariables(X=1.0,Y=0.5,Z=0.5)whilewiththe revised values (after the first iteration) of X=4.25, Y=0.2 and Z=0.15avalueof8.795105_m3_{/kWhasbeenachieved.Table2}

showstheimprovementoftheobjectivefunctionV=Baftereach stepoftheoptimizationaswellasthecombinationofslowmoving variablesthatyieldedthebestresult.

Thenewfoundvaluesfortheslowmovingvariablesareusedas startingpointforaseconditeration.Interestingly,thevolumeof materialperunitofexergytransferdecreasescontinuouslyasX increases until reaching very small values. After X=11 the LP stream presents a very small Reynolds number for being consideredasaturbulentflow(<4000)[26]duetoaconsiderable decreaseinthevelocityoftheair.Aturbulentflowisdesiredin ordertomaintainahealthyheattransferinthepipes,thereforethe sweepisstoppedatX=11.SubsequentsweepsweremadeforY and Zandnochangewasfoundintheirvalues,thustheoptimization loopwasconcluded.

Thefinaldesignforthecross-sectionalareaofthesegmentof theHXisobtainedaftertheslowmovingvariableshavebeenfine tuned. The geometric parameters and relevant performance metrics ofthedesign generatedfor thecasestudyaregiven in Table3.

A volume of 8.485105m3 of material per kW of exergy transferwithanexergyefficiencyof99%isachieved.Forreference, aconventionalshell-and-tubeHXthatoperatesbetween820and 350K on the tube side with a pressure of 7MPa requires of approximately14.917103_m3_{perkWofexergytransferwithan}

efficiencyof95%.Thevolumeobtainedattheendofthe one-factor-at-a-timeoptimizationtranslatesroughlyintoacostof53s/kW (duetomaterialonly)consideringanapproximatecostof80_s/kg forthesteelpowderforlasermanufacturing[27].Althoughthe materialused representsa largeshare ofthetotalcost ofa 3D printedpart;additionalfabricationcostssuchaslabour(e.g. pre-processingofthejob)and–specially–machinetimearebyno meansnegligible.

Thecurrentcostingmethodsusedforthefabricationofone-off andshortruncomponentsarebasedonpre-establishedratesper unitvolumethathavealreadyfactoredintheadditionalfabrication

Distance between centres of HP Pipes (m)

0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275

Temperature Gradient (K/m)

10-1 100 101 102

T=400 W=99% T=400 W=98% T=400 W=97% T=745 W=99% T=745 W=98% T=745 W=97%

Fig.12.TemperaturegradientthestreamsfordifferentvaluesofD.

Yes

Find new value

for X us

ing same

Y and Z

Find new value

for Y us

ing new X

and same Z

Find new value

for Z usi

ng new X

and new Y

Values

changed ?

No

Table2

Reductioninvolumeperunitofexergytransferaftereachstepoftheoptimization.

Iteration Slowmoving variablechanging

X Y Z V=B(m3_/kW)

0 N/A 1 0.5 0.5 1.716104

1 X 4.25 0.5 0.5 1.669104

Y 4.25 0.2 0.5 1.280104

Z 4.25 0.2 0.15 8.795105

2 X 11 0.2 0.15 8.485105

costs;sotheintricacyofthedesigndoesnothavearealimpacton theendcostofthecomponent.Itisnoteworthythatafurthercost reductioncanbeachievedforholloworperforateddesignsthat allowdifferentcomponentstobebuiltsimultaneouslywithintheir cavitiesthusmaking amore efficient useofmachine'srunning time.Both,materialandindirectcostsareexpectedtodecreaseas additive-manufacturingtechnologiesmature.Thestandard addi-tive manufacturing equipment commonly found nowadays can fabricatecomponentswithwallthicknessesdowntoonetenthofa millimetre,which rendersmostofthedesignsevaluatedinthe studyasnon-fabricable;howeverthecapabilitiesofthemachinery arerapidly improvingandcurrent state-of-the-artequipment is already capable of fabricating elaborate micro-scale structures [28].

Itisimportanttoemphasizethatthedesignanalyzedasacase studyisjustanexampleusedtodemonstratethepremisethatitis possible to create designs for heat exchangers that are highly exergy-efficient and very cheap (due to the small volume of materialrequired)iftheconstrainsimposedbythelimitationsof traditionalmanufacturingmethodsaresetaside.

5.Concludingremarks

Various new manufacturing techniques, such as additive manufacturing, have been developed in recent years; these methodsallowthefabricationofintricatedesignswhose fabrica-tionwould notbe possibleorcost-effective throughtraditional methods.

A design for the cross-sectional area of a gas to gas heat exchangerbasedonanhexagonalmeshhasbeenproposedwith whichaconsiderablevolume(hencecost)reductionperunitof exergy transfer can be attained. This geometric configuration wouldbehighlyimpracticaltomanufactureinaconventionalway butcouldbeeasilybuiltbymeansofmoderntechniques.

Asegmentoftheheatexchangerthatwillworkatanaverage temperatureof745Kwasdesignedandoptimizedandavolumeof steelaslowas84.846cm3_{perkWofexergytransferata99%exergy}

efficiencywasobtained.Suchgeometrycanbefabricatedcurrently atcompetitivecosts;andafurtherreductionincostisexpectedas thenewmanufacturingmethodsmature.

Thestudyhasrevealedaveryimportantfact.Thevolumeper kWof exergytransfer, which is oneof thecost-drivingfactors, increases in nearly quadratic proportion with respect to the characteristic dimension (distance between centres of high pressure pipes) of the heat exchanger. This means that if the geometryisscaledupbydoublingthedistancebetweenHPpipes, the volume of material per kWof exergy transfer will not be doubled–asintuitionwouldsuggest–butitwillnearlyquadruple.

Theaforementionedobservationmaybetrueforothertypesof heatexchangersandwithfurtheranalysisitcouldberegardedasa rule of thumb. It strongly suggests that the design of heat exchangersshouldaimtowardssuchasmallscaleaspossiblefor substantiallyincreasingperformanceandreducingcosts.

Acknowledgements

This work is an outcome of the UK-China NexGen-TEST collaborative research project; which has been funded by the UKEngineeringand PhysicalSciences ResearchCouncil(EPSRC) and the National Natural Science Foundation of China (NSFC). Special thanks to the organizing committee of the OSES 2016 ConferenceinValletta,Maltainwhichthisworkwaspresented.

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GeometricparametersandperformancemetricsofthefinaldesignoftheHX segment.

Operationalparameters Geometric parameters

Performanceparameters

X 11.0 – D 0.01 m W 0.99 –

Y 0.2 – rHP 5.701E4 m DPHP 2423.785 Pa/m Z 0.15 – tHP 2.699E5 m DPLP 4.4649 Pa/m PHP 5.0 MPa tLP 3.958E6 m rT 44.414 K/m PLP 101.325 kPa L 4.403E3 m V=B 8.485105

Tavg 745 K Ar 0.0239 –

THP 749.967 K l 0.0063 –

TLP 740.032 K

_

mHP 6.494105 kg/s

_

mLP 3.247105 kg/s UHP 2.739 m/s ULP 1.593 m/s

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