NH3 condensation in a plate heat exchanger

NH3 condensation in a plate heat exchanger

Flow pattern based models of heat transfer and frictional pressure drop

Tao, Xuan; Infante Ferreira, Carlos

DOI

10.1016/j.ijheatmasstransfer.2020.119774

Publication date

2020

Document Version

Final published version

Published in

International Journal of Heat and Mass Transfer

Citation (APA)

Tao, X., & Infante Ferreira, C. (2020). NH3 condensation in a plate heat exchanger: Flow pattern based

models of heat transfer and frictional pressure drop. International Journal of Heat and Mass Transfer, 154,

[119774]. https://doi.org/10.1016/j.ijheatmasstransfer.2020.119774

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

ContentslistsavailableatScienceDirect

International

Journal

of

Heat

and

Mass

Transfer

journalhomepage:www.elsevier.com/locate/hmt

NH

₃

condensation

in

a

plate

heat

exchanger:

Flow

pattern

based

models

of

heat

transfer

and

frictional

pressure

drop

Xuan

Tao

a,∗

_,

_Carlos

_A.

_Infante

_Ferreira

a

a Process and Energy Laboratory, Delft University of Technology, Leeghwaterstraat 39, 2628 CB Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 27 January 2020 Revised 6 April 2020 Accepted 7 April 2020 Keywords: Condensation mechanism

Heat transfer model based on flow pattern Lockhart and Martinelli model

Separated flow Plate heat exchanger NH 3

a

b

s

t

r

a

c

t

ThispaperdevelopspredictingmodelsforNH3condensationinplateheatexchangersbasedonthe

exper-imentsofflowpatterns,heattransfercoefficientsandfrictionalpressuredroppreviouslyreportedbythe authors.TheaimistoprovidedesignmethodsofcompactplatecondensersusedinNH3systems,which

arenotavailableinopenliterature.Theexperimentaldataarefirstlycomparedwithselectedcorrelations, showingapooragreement.Aheattransfermodelisdevelopedbasedonflowpatterns,whichrepresents the transitionfrom convective condensationtogravity-controlled condensation. The physical interpre-tation ofthetwo-phasemultiplier approachand the deviationfrom Nusselt’stheoryarediscussed. A transitioncriterionofcondensationmechanismsisproposedbasedonthewettingcharacteristics.Since theflowpatternsindicateseparatedflow,theLockhartandMartinellimodelisselectedandismodified topredictthe frictionalpressure drop.Themodel isthesum oftheliquidpressuredrop,vapor pres-suredropandinterfacepressure drop.Thecontributionsofvaporpressuredropandinterfacepressure droparediscussedandquantified.Theproposedheattransferandfrictionalpressuredropmodelsshow goodpredictiveperformances.NH3flowhaslargetwo-phaseslipbecauseofthelargedensityratio.Plate

heatexchangershavecorrugatedchannelsandtendtobreakuptheliquidfilm.Themodelsidentifythe distinctflowcharacteristicsbasedonflowpatterns.

© 2020TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

NH3 is an environmentfriendly naturalrefrigerantwith supe-rior thermalproperties.However, itsapplicationisrestrained due to safety issues. Plate heat exchangers (PHEs) have the potential to beusedin NH3 systems suchasfortherecovery oflow grade heat.The compactstructures areable totransferlarge heatloads withreducedchargeofworkingfluid.Thusthesafetyrisk canbe mitigated.Furthermore,PHEshavetheadvantage ofhighthermal effectivenessanddesignflexibility,bringingaboutwideutilization inrefrigeration,pharmacyandchemical engineering.PHEsconsist of outside frame plates and inside stacked plates. Stacked plates usually havesinusoidalcorrugationsandprovideheat transfer ar-eas.Twoadjacentplatesformflow channelswhosegeometryand areachangeperiodically[1,2].

Two-phase vertical downward flow in PHEs has been widely experimentally investigated, which isthe common flow direction of condensers and absorbers [3–5]. Tao et al. [2] reviewed the

∗ _{Corresponding Author.}

E-mail addresses: x.tao@tudelft.nl (X. Tao), c.a.infanteferreira@tudelft.nl (C.A.I. Ferreira).

condensation mechanisms in PHEs. The transition from gravity-controlledcondensationtoconvectivecondensationisdetermined by mass fluxes and vapor qualities. Larger mass fluxes promote convective condensation. Furthermore, the fluid properties play important roles. The flow tends to be separated when the two-phase density ratio is large, and is close to homogeneous flow withsmalldensityratio.Condensation pressures,inlet superheat-ing,plategeometriesandtemperaturedrivingforcesalsoinfluence theheattransferandfrictionalpressuredrop[2].

Tao and Infante Ferreira [6] reviewed the heat transfer and frictionalpressuredropcorrelationsforcondensation inPHEs.An experimental database is developed including 2376 heat transfer dataand1590frictionalpressuredropdata.Theworkingfluidsare HFCs,hydrocarbons,HFOsandCO2,butNH3 isnotincluded.Most correlationshavebeenderivedfromexperimentaldata,andare ap-plicableintheoriginalandsimilaroperatingranges.Inordertobe assessed inlarger ranges,thesecorrelations havebeencompared withtheexperimental database. The heat transfer correlationsof Longo etal. [4]andKuo etal.[7] predict the experimental data best.Anewcorrelationisproposedforfrictionalpressuredrop[6]. ExperimentaldataofNH3 condensationinPHEsarescarce,and details of the experiments are missing in old papers [8,9].

Re-https://doi.org/10.1016/j.ijheatmasstransfer.2020.119774

Nomenclature

Symbols

Co Convectionnumber[-]

d Diameter[m]

dh Hydraulicdiameter[m] f Darcyfrictionfactor[-]

Fr Froudenumber[-]

g Gravitationalconstant[ms−2]

G Massflux[kgm−2s−1]

h Enthalpy[Jkg−1]

jG Non-dimensionalgasvelocityEq.(1)[-] Lp Port-to-portplatelength[m]

P Pressure[Pa] Pr Prandtlnumber[-] Re Reynoldsnumber[-] T Temperature[°C] v Superficialvelocity[ms−1] We Webernumber[-] x Vaporquality[-] X Lockhart-Martinelliparameter[-] Greeksymbols

α

Heattransfercoefficient[Wm−2 K−1]

β

Chevronangletoflowdirection[°]



Difference[-] ɛ Voidfraction[-]

φ

L Two-phasemultiplier[-]



FractionofconvectivecondensationinEq.(16)[-]

λ

Thermalconductivity[Wm−1 K−1]

μ

Dynamicviscosity[Pas]

ρ

Density[kgm−3]

σ

Surfacetension[Nm−1] Subscripts av Averaged c Combinedcondensation cc Convectivecondensation cr Criticalcondition

exp Experimentaldata

G Gasorvapor

gc Gravity-controlledcondensation

GO Gasorvaporonly

int Attwo-phaseinterface

L Liquid

LG Latentliquidtovapor

ll Laminar-laminarflow

LO Liquidonly

LT1 Limit1inEq.(8)

LT2 Limit2inEq.(9)

pre Predicteddata

sat Atsaturationconditions

TP Two-phase

T Transitionvalue

tt Turbulent-turbulentflow

wall Atwallconditions

cently,the authorsexperimentally investigated the flow patterns, heattransfercoefficients(HTCs)andfrictional pressuredrop[10]. NH3hasdistinctfluidpropertiesfromHFCsandhydrocarbons.NH3 ischaracterizedbylargetwo-phasedensityratio,thermal conduc-tivity, surface tension and latent heat. Consequently, the experi-mental HTCs and frictional pressure drop show sharp sensitivity tothevaporqualities[10].Tothebestoftheauthors’knowledge, nocorrelation hasbeen speciallyproposed forNH3 condensation

in PHEs. In tubes, the correlations derived from HFCs or hydro-carbons mismatch the experimental data of NH3 [11,12]. The ac-curacyofcrosscalculationisunknowninPHEs.Thiswork investi-gates NH3 condensationinPHEs.Derivedfromexperimental data

[10],thispaperdevelopsheattransferandfrictionalpressuredrop modelsbasedonflowpatterns.Theprimaryphysicalprocessesare described,whiletheflowdetailscannotbequantifiedandare con-sideredbyinvolvingempiricalconstants.

2. Previousphysicallybasedmodelingofcondensation

Duringcondensation intubes, the theory ofheat transferand frictionalpressuredrophasbeenestablishedtodescribethe phys-icalprocessoftwo-phaseflow.ThetheoreticalresearchesonPHEs arelimited, andthegeometric influenceisnot thoroughly under-stood. ThisSection reviewsthe previous studies that are usedas thestartingpointstodevelopthepredictingmodels.

2.1. Heattransfermodelsbasedonflowpatterns

Duringthecondensationinhorizontaltubes,flowpatternbased heat transfer models have been recognized to be accurate and widely applicable. Annular flow is modelled as fully convective condensation. Stratified flow is considered to combine gravity-controlledcondensationatthetopofthetubeandconvective con-densation at the bottom, and gravity-controlled condensation is usuallydominant.Slugandchurnflowsfallintothesetworegimes andare not analyzedseparately. Bubbly flow only takes placeat large mass fluxes, whichgo beyondthe generaloperating ranges ofcondensation[13–16].Somepapersuseddifferentterms,butthe physicalinterpretationissimilar[17].

DobsonandChato[14]experimentallyanalyzedtherelianceof HTCsonflowpatterns.Theflowpatternsareclassifiedinto gravity-controlled flow and shear-controlled flow. In gravity-controlled regime, HTCs are sensitive to temperature driving force but are almost independent ofmass fluxes. The HTCs ofshear-controlled flowarecharacterized bythesignificantinfluences ofmassfluxes and vapor qualities. These authors proposed mechanistic models forthetworegimes.

Thome et al. [16] assumed uniform liquid film thickness for annular flow. The film thickness is calculated using a void frac-tion model. Convective condensation takes place during annular flow.Forstratifiedflow,thefilmisconsideredasapartofthe an-nular ring located atthe bottom, where convective condensation prevails.The wall isunwettedatthe top,wherethe Nusselt the-oryis applied.Theportionsofthetwo condensationmechanisms are determined by stratified angles. In order to predict NH3 in-tube(8mmdiameter)condensation,ParkandHrnjak[12]keptthe flowpatternmap andcondensation mechanismsofElHajal etal.

[15]andThomeetal.[16],butmodifiedtheheattransfer correla-tions.

Cavallini et al.[13] divided the condensation mechanisms ac-cordingtothedependenceontemperaturedriving force. Temper-aturedrivingforceindependentregimeisequivalenttoconvective condensation, while temperaturedriving force dependent regime is similar to gravity-controlled condensation. In Eq. (1), jG is the

non-dimensionalgasvelocity.WhenjGislargerthanthetransition

value in Eq. (2), convective condensation applies. CT depends on

the working fluids. 1.6 is recommended for hydrocarbons, while 2.6 applies forother fluidssuch asHFCs, NH3, CO2 and H2O. At smaller values of jG, the HTC combines convective condensation

andNusseltcorrelations. Fronk andGarimella[11,18]developed a heattransfermodelforNH3condensationinsmalldiametertubes (0.98,1.44and2.16mm).TheyalsousedjGasthetransition

crite-rion.Duringannularflow,theheattransferisenhancedbythe in-terfacialroughnessarising fromtwo-phase momentumdifference.

Fig. 1. Transition of condensation mechanisms. The criteria of Cavallini et al. [13] , El Hajal et al. [15] and Dobson and Chato [14] were developed for horizontal tubes. Shah [17] included horizontal, vertical and inclined tubes.

ThecorrelationsofCavallinietal.[13]wererecommendedfor non-annularflow. jG= xG [gdh

ρ

G

(

ρ

L−

ρ

G

)

]0.5 (1) jG,T=

⎡

⎢

⎣

⎛

⎜

⎝

7.5 4.3



μL μG

0.1

_ρG ρL

0.5

_1−x x

0.9



1.11 +1

⎞

⎟

⎠

−3 +C_T−3

⎤

⎥

⎦

−0.333 (2)

The above models have been proposed for horizontal flow. Shah [17] specified the influence of flow directions. During hor-izontal flow, condensation mechanisms were divided into con-vective condensation and gravity-controlled condensation. Fully gravity-controlled condensation is only included for vertical and inclined flow. The difference between gravity-controlled conden-sationandfullygravity-controlled condensationis theconvection effect. For gravity-controlled condensation, convection still con-tributes to heat transfer. The convection attenuates during fully gravity-controlledcondensation.Insteadofanalyzingflowpatterns, Shah [17] developed the heat transfer models by fitting a large numberofexperimentaldata.

Fig. 1 shows several transition lines. Dobson and Chato

[14]used Soliman’s modifiedFroude numberto distinguish strat-ifiedflow from annularflow. The transitiongenerallyhappens at highervaporquality forsmallmassfluxes.El Hajaletal.[15] dif-ferentiated stratifiedflow andannularor intermittentflow using thevoidfractionandstratifiedangle.Stratifiedflowincludes fully-stratified flow and stratified-wavy flow, whose difference results from the occurrence of waves at the two-phase interface. These two sub-flow patterns are considered the samein terms of con-densationmechanisms.Cavallinietal.[13]classifiedthe condensa-tionmechanismsbyreferringtoEqs.(1)–(2).ThevariationofHTCs is gradual duringtransition. According to Shah [17], vertical and inclinedflowshavesmallertransitionmassfluxesthanhorizontal flow. Fully gravity-controlled condensationonly happensatsmall massfluxesandhighvaporqualitiesforverticalandinclinedflow,

where little condensate accumulates at the bottom of the tubes andconvectionisnegligible.

2.2.Frictionalpressuredropmodelsbasedonflowpatterns

Two-phase flow is divided into homogeneous flow and sepa-rated flow. The homogeneous model assumes the two-phase ve-locitiesarethe same,andtheflow isconsideredasanequivalent fluid.Bubbly flow andmist flow aregenerally categorizedas ho-mogeneousflow.Accordingtotheseparatedflowmodel,vapor ve-locityislargerthanliquidmainlybecauseofthedensityratio. Sep-aratedflowincludesstratifiedflowandannularflow.

The separated flow model calculates the two-phase frictional pressuredropbysummingtheliquidpressuredrop,vaporpressure drop and the pressure drop at the interface. Lockhart and Mar-tinelli [19] analyzed separated flow by assuming the static pres-suredrops ofliquidandvaporarethe same.Thisassumption ap-pliestospaceunchangedflow patternsandexcludesintermittent flow. Chisholm[20] developedthetheoretical basis ofthismodel andspecifiedtheinterfacialshearforce. Accordingtothe theoret-icalassumption,theliquidandvaporstreamshavethesameflow mechanisms duringtwo-phase flow as single-phase flow. Homo-geneous flow is a special caseof this model with zero slip and uniformdensity.Inthiscase, theexperimental dataare generally over-predicted.

Friedel[21] claimedthat the frictional pressure dropdepends onthe flowdirectionbecausethe slipatthetwo-phase interface isdifferent.Undergravity,liquidmovesfasterverticallydownward, resultinginlargervoidfractionorsmallerslipratio.One correla-tionwasproposedforhorizontalandverticallyupwardflow,while anotheronewasdevelopedforverticallydownwardflow. Addition-ally,surfacetension affects thefrictional pressure dropby acting onthetwo-phaseinterface.

Müller-Steinhagen and Heck [22] analyzed the sensitivity of frictionalpressuredropwithvaporqualities,andobservedapeak value for x≈ 0.85. At higher vapor qualities,the frictional pres-suredropdecreasesasasmalleramountofliquidreducesthe in-terfacialshear force. A correlation was developed and was com-paredwithothers.Thiscorrelationsatisfactorilypredictedthe ex-perimentaldatabase which is mainly composed ofair-water and steam-water.The homogeneous correlationswere only applicable forlow vaporqualities, whileLockhartand Martinelli[19]model generallyover-predictedtheexperimentaldata.

DuringNH3 condensationinatube(8mmdiameter),Parkand Hrnjak[12]recommendedthemodelsofFriedel[21]andMüller– SteinhagenandHeck [22] for therange above 1kPa m−1, which werepredictedto beintermittent orannularflow.Forvalues be-low1kPa m−1,thehomogeneousmodelhadbetterperformance. Under these conditions stratified flow was expected to occur. It is separated flow, but the two-phase velocity difference is small becauseofthe smallmassfluxes. Fronk andGarimella[23] mea-sured the frictional pressure drop during NH3 condensation in-sideasmalldiametertube(1.44mm).Theexperimentaldatawere under-predictedby theFriedel[21]modelforabout30%.The de-viationismainlyattributedtothelargesurfacetensionofNH3.

Fig.2comparestheseparatedflowmodels.Thetrendsofthese modelsaresimilar. Withincreasingvaporqualities,thetwo-phase pressuredropincreasesatlowandintermediate values,andthen decreaseswhenapproachingpurevapor.Themaximumtwo-phase pressuredropisbetweenx₌0_.8 andx₌0_.9.Thechanges ofthe liquidandvaporpressuredroparemonotonic.Vaporpressuredrop is much larger than liquid pressure drop because the two-phase densitiesandviscositiesofNH3 differdramatically.Theinterfacial pressuredropislargeratintermediate vaporqualitieswhenboth phasesplayroles.

Fig. 2. Comparison of models by dividing the two-phase pressure drop into the liquid pressure drop, vapor pressure drop and the pressure drop at the interface. The separated models are Lockhart and Martinelli [19] model (L&M) [20] , Friedel [21] model and Müller–Steinhagen and Heck [22] model (M-S&H). The single-phase correlation is from Kast et al. [49] .

3. Comparisonwithexistingcorrelations

The authors have reported experimental HTCs and frictional pressure drops of NH3 condensation in a PHE [10]. The experi-ments coverthe mass fluxesof 21–78 kg m−2 s−1, theaveraged vapor qualities of 0.05–0.65 and the saturated pressure of 630– 930kPa. Inthetestedranges,the flowpatternsarefull filmflow andpartialfilmflow. HTCsincrease significantlywithvapor qual-itiesandarelesssensitivetomassfluxes.Frictionalpressuredrop increasessharply withboth vapor qualities andmass fluxes. The HTCsandfrictionalpressuredropshowthecharacteristicsof sep-aratedflow. The flow patternsresult fromthefluid properties of NH3. The experimental data are compared withselected correla-tionsinthisSection.Thecomparisonispresentedintermsofmass fluxesandvaporqualities.

3.1.Heattransfercorrelations

Tao and Infante Ferreira [6] summarized condensation heat transfercorrelations in PHEsand developed an extensive experi-mentaldatabase.Eightcorrelationshavebeenassessedby compar-ingwiththedatabase.ThecorrelationsofLongoetal.[4]andKuo etal.[7]showthebestperformance.Nevertheless,NH3 isnot in-cludedinthedatabase.

Fig. 3 compares thesetwo correlations withthe experimental data for three mass fluxes. Kuo et al. [7]’s correlation is a two-phasemultiplier approach,whichassumesannularflow [24].The condensationheattransferissimilartotheconvectiveheat trans-fer of liquid phase since all the heat is transferred through the liquid[24].Thevapor flowandheatflux enhancetheheat trans-fer and are considered as a two-phase multiplier. According to thiscorrelation,HTCsincreasesignificantlywiththevaporquality, agreeingwiththetrendoftheexperimentaldata.

The correlation of Longo et al. [4] is composed of convective condensationandgravity-controlled condensation.The convective correlationinvolvestheequivalentReynoldsnumber,whichtreats thetwo-phaseflowasasingleequivalentfluid.Thevaporflow is replacedwithanadditionalliquidflowbykeepingthesameshear forceatthetwo-phaseinterface.Theconversionratiodependson thetwo-phasedensityratio[25].Thesensitivitytothevapor

qual-Fig. 3. Condensation HTCs of NH 3 with varying averaged vapor quality and mass fluxes. Comparison of experimental and predicted data. The prediction is from Longo et al. [4] (solid lines) and Kuo et al. [7] (dashed lines).

Fig. 4. Comparison of NH 3 condensation HTCs with correlations. Mean error (ME): 1

n  n

1α

pre−αexp

αexp ; Mean absolute error (MAE):

1 n

 n 1|α

pre−αexp|

αexp ; Root mean squared er-

ror (RMSE): 1 n

 n 1(αpreα−expαexp)

2

; Per ± 30% : Percentage of experimental data within ± 30%; Per ± 50% : Percentage of experimental data within ± 50%.

ity is less noticeable than for the experimental data. Convective condensation is transformed into gravity-controlled condensation atlow vapor qualities, andthe HTCs become insensitiveto mass fluxesandvaporqualities.

Fig.4presentstheaccuracyofthecorrelations. Thecorrelation of Kuo etal. [7] predicts 75.7% of the experimental data within ±30%. The meanabsolute error (MAE)is 21.3%. It estimates cor-rectlythesignificantsensitivitytovaporqualitiesandshows con-sistent trend withthe experimental data. Nevertheless,predicted dataofdifferentmassfluxesareseparated.Thiscorrelationismore accurate forintermediateandlarge massfluxes, anddramatically under-predictsthedataofsmallmassfluxes.Longoetal.[4]’s cor-relation predicts 72.4% ofthe data within ±30%. It over-predicts theexperimental HTCs atlow vaporqualities, andunder-predicts theexperimental HTCsathighvapor qualities.Thepredicted val-uesoflargeandintermediatemassfluxesconverge,butthe under-predictionforsmallmassflux isnoticeable.In short,the

correla-Fig. 5. Frictional pressure drop of NH 3 with varying averaged vapor quality and mass fluxes. Comparison of experimental and predicted data. The prediction is from Tao and Infante Ferreira [6] and is based on the void fraction of Zivi [33] (solid lines) and Rouhani and Axelsson [34] (R&A) (dashed lines).

tion ofKuo etal.[7]over-estimatesthe influenceof massfluxes, whileLongoetal.[4]’scorrelationcannotpredictthesharp sensi-tivitytovaporqualities.

3.2. Two-phaseFanningfrictionfactorcombinedwithvoidfraction

Tao and Infante Ferreira [6] also assessed six frictional pres-sure drop correlations with an experimental database, including two-phase Fanningfriction factor andLockhart–Martinelli model. The agreement ofthe correlationsispoor, anda newcorrelation hasbeendeveloped.InthepreviousresearchesofPHEs,Lockhart– Martinelli’smodelismostlychosenforair-waterflow[26–28].The air-watersystemischaracterizedbyalargetwo-phasedensity ra-tio,andseparatedflowpatternssuch asfull filmflowandpartial filmflowcoverlargeranges[2].Thedatabaseismostlycomposed of HFCs, hydrocarbons and HFOs, whosetwo-phase densityratio isrelatively small[6].Nino etal.[29] andAdams etal.[30] sug-gested the homogeneous void fraction for fluidsof small liquid-vapordensityratioandseparatedvoidfractionforlargedensity ra-tio.Thenewcorrelationassumeshomogeneousflowandcalculates thetwo-phase Fanningfriction factor.WhenappliedforNH3,this correlation significantly over-predicts the experimental data [31]. The two-phase slip ratio of NH3 is much larger than 1. The ho-mogeneousvoidfractionoverestimatestheaveragedvelocity and thus the shear force. Jassim et al.[32] proposed to calculatethe averageddensityusingtheseparatedvoidfractionmodel.

ThevisualizationexperimentsindicatethattheNH3flowis sep-arated [10].To the bestof the authors’knowledge, no void frac-tion model is speciallydeveloped forPHEs. Tao et al. [31] com-paredseveralvoid fractionmodelsoriginally proposed for micro-channels.InFig.5,themodelsofZivi[33]andRouhaniand Axels-son [34]are usedtocalculatetheaverageddensity.Zivi[33] pro-posedatheoretical modelandclaimedtoprovidethelower limit ofvoidfraction.RouhaniandAxelsson’s[34]modelwasoriginally derivedforflowboiling,andismoresuitableforlowpressure.This modelischosenbecausethecondensationinthispaperhappensat relativelylowpressure.

According to Fig.5, both predictionmethods show larger val-uesthantheexperimentaldata.Thepossiblereasonisthatthe es-timated voidfraction islarger than theactual values inPHEs.At lowvaporqualities,thepredictionbasedonZivi[33]voidfraction

Fig. 6. Influence of geometrical structure on void fraction. Round tubes and oval tubes with and without microfins are compared. The oval tubes with 0.974 mm height are flatter than those with 2.57 mm (adapted from [35] ).

hasdeviationslargerthanthreetimes.Theslopeofthelinesbased onRouhaniandAxelsson’s[34]voidfractionbecomesflatat inter-mediatevaporqualities.TheflowpassageofPHEsisapproximately rectangular,andtheratioofwidthtolengthissmall.Thechannel hasacorrugatedstructureandisinterruptedbythecontactpoints ofadjacentplates.Thevoidfractionisdifferentfromwhatapplies forsmoothchannels.Liquidislikelytobeheldatthecornerofthe flowpassagesandatthecontactpoints, whichreducestheliquid velocityandincreasestheslipratio.Thevoidfractionconsequently decreases.Fig.6showsthevoidfractioncalculatedbyWilsonetal.

[35]’smodel,whichindicatesthegeometryinfluencebycomparing roundtubesandovaltubeswithandwithoutmicrofins.Thevoid fractioninovaltubesislowerthaninroundtubes.Theovaltubes are flatter for smaller heights and have lower void fraction. Mi-crofintubeshavegenerallylowervoidfractionthansmoothtubes. Thevoidfractionofsmooth-roundtubesisclosetoZivi[33]model, whilethemicro-finsandovalstructures reduce thevoidfraction. Moreover, in Fig. 5, the experimental data increase sharply with vapor qualities, which is different fromthe trend of the predic-tions.As showninFig.2,thesharp increaseisa characteristicof separatedflow.

Althoughthe calculationproposed by Taoand InfanteFerreira

[6]hasbeenderivedfromanextensiveexperimentaldatabase,the applicationislimitedtotheworkingfluidscoveredbythedatabase (HFCs,hydrocarbonsandHFOs),whichare characterizedby small two-phasedensityratioandtendtostreaminhomogeneousflow. Triplett et al. [36] concluded that the homogeneous model pre-dicted correctly the frictional pressure drop of bubbly and slug flow,butsignificantlyover-estimatedthedataofannularflow.The flowofNH3 isseparated.Thefrictionalpressuredropneedstobe analyzedmakinguseofadifferentapproach.

4. Developmentofaheattransfermodel

ThisSectionfirstlydevelopsaheattransfercorrelationfor con-vectivecondensation,andthen establishesthetransitioncriterion of condensation mechanisms depending on the wetting charac-teristics.Aheat transfercorrelation forcombinedcondensation is alsodeveloped, andis composed ofconvective condensation and gravity-controlledcondensation.Theexperimentaldatareportedin Taoetal.[10]areusedasthebasisforthesecorrelations.

4.1.Convectivecondensation

During NH3 condensationin a PHE, the flow patternsare full filmflowandpartialfilmflowbelow100kgm−2 s−1[10].Forfull filmflow, thewallsurfaceiscompletelywettedbytheliquid.Full filmflowisequivalenttoannularflowintubes,buttheliquid can-notdeveloparegularringaroundtheplatesurfaceduetothe cor-rugated flow passages. The interaction between the vapor inthe coreandtheliquidarounditgivesrisetoconvectivecondensation. All the heat is transferred through the liquid film, wherethe heattransfer processis similarto single-phaseflow. Vaporphase shavesthetwo-phaseinterfaceandenhancestheheattransfer.The two-phasemultiplierapproachwasoriginallydevelopedfor annu-larflow intubes and isapplicable for thefull filmflow in PHEs

[24]. The two-phase multiplier depends on the vapor flow and two-phasefluidproperties.ItgenerallyhastheformofEq.(3).

α

cc

isthe HTC of convective condensation.

α

L isthe liquid HTC and

onlyidentifiestheliquidmassflux.Itiscalculatedusingtheliquid Reynoldsnumber,ReL.InEq.(4),theliquidonlyHTC,

α

LO,assumes

allthefluidisliquidandisdeterminedaccordingtoVDI[1].fLO is

theDarcyfrictionfactorandiscalculatedinEqs.(8)–(10)[1].This equationis derived fromalarge rangeof geometricalparameters and agrees well with the water HTCs determined previously by theauthors [37,10].Eq.(5) showsthedifference between

α

L and

α

LO. fL/fLO is approximately1 especially forlarge chevron angles. Eqs.(6)–(7)calculateReLOandReL.

α

cc=

α

LF

(

x,

ρ

L/

ρ

G,G

)

(3)

α

LO=0.122

(

fLOsin2

β

)

0.374Re0LO.748Pr0L.333

 μ

_μ

wall



0.167

λ

L dh (4)

α

L=0.122

(

fLsin2

β

)

0.374Re0L.748Pr0L.333

_μ μwall

0.167_λ L dh =

α

LO

fL fLO

0.374

(

1− x

)

0.748_≈

_α

LO

(

1− x

)

0.748 (5) ReLO= Gdh

μ

L (6) ReL= GLdh

μ

L = G

(

1− x

)

dh

μ

L (7) fLT1=



64Re−1, Re<2000

(

1.8lg

(

Re

)

− 1.5

)

−2, Re≥ 2000 (8) fLT2=



3.8

597Re−1+3.85

, Re<2000 3.8

39Re−0.289

_{, Re≥ 2000} (9) f=



cos

β



0.18tan

β

+0.36sin

β

+fLT1/cos

β

+1



−cos

β

fLT2



−2

(10)

In order to confine the experimental data to fully developed convectivecondensation,only datawithGL >40 kgm−2 s−1 are

includedtodevelop thecorrelation.Eq.(11)isobtainedby multi-variableregressionanalysis.The firsttermofthetwo-phase mul-tiplierinterprets theenhancement contributedbythevapor flow. Itapproaches0whenxis0,indicatingthattheenhancement van-ishesasthefluid becomesliquidonly. InEq.(12),Co is the con-vectionnumberandrepresentstheslipvelocityatthetwo-phase interface.Asthe reducedpressureincreases,thetwo-phase prop-ertiesbecomealikeandthedensityratioiscloseto1,suppressing theslipattheinterface.InEq.(13),theliquidFroudenumber,FrL,

isthe ratio of inertia to gravity, andindicates the dominance of momentumeffectorstratifyingeffectforseparatedflow.The sec-ondterminthebracketofEq.(11)specifiesthedifferencebetween

Fig. 7. Applicability of the correlation of convective condensation ( Eq. (11) ) in terms of We L .

α

LOand

α

L.

α

ccbecomes

α

LO whentheflowissingle-phaseliquid.

α

cc=

α

LO

0.17Co−1.12_Fr−0.2 L +

(

1− x

)

0.748

₍₁₁₎ Co=

ρ

G

ρ

L



0.5



_{1− x} x



0.8 (12) FrL= G2

ρ

L2gdh (13)

The previous analysis is restricted to the data with

GL > 40 kg m−2 s−1. Convective condensation extends to lower

liquid massfluxes. In smalldiameter channels,the magnitudeof surface tensionbecomes prominent relative to gravity and shear force. Larger surface tension promotes the change from annular flow to wavy flow in tubes [38]. The surface tension affects the wetting characteristics and condensation mechanisms, while the inertial force tends to distributethe liquid film around the wall surface. InEq.(14),theliquidWeber number,WeL,istheratioof

liquid inertia to surface tension. WeL is used to distinguish the

condensation mechanisms. Eq.(11) is used to predict all the ex-perimentaldata,andthecomparisonispresentedinFig.7.When

WeL > 0.12, the experimental data are well predicted, and the

deviation is within _±20%. As WeL < 0.12, the experimental data

areunderpredicted.Anothercondensationmechanismisinvolved, whichenhancestheheattransfer.ThevalueofWe_L,T=0.12isthe transitioncriterionofcondensationmechanisms.

WeL=

ρ

L

v

2 Ldh

σ

= G2

₍

_{1− x}

₎

2_d h

ρ

L

σ

(14)

ThetransitionlineispresentedinFig.8,whichagreeswellwith thechange offlowpatterns. Ascompared withFig.1,the transi-tionmass fluxissmaller thaninhorizontaltubes, butiscloseto thevalueforverticalandinclinedtubes.Theflowdirectionaffects the transition ofcondensation mechanisms because ofthe inter-actionbetweengravityandshearforce.Forfullygravity-controlled condensation,thecondensatefilmformedonthewallflowsalmost verticallydownward.Bycontrast,interfacialshearforcedominates for convective condensation. The condensatefilm is less affected by gravity and flows along the main flow direction. The transi-tiondependsontherelativemagnitudesofgravityandshearforce.

Fig. 8. Comparison of the flow pattern map with the criterion of condensation mechanisms ( W e L,T = 0 . 12 ).

The flow inthe corrugated groove of PHEs is similar to inclined downwardflowintubes[10].Thegravityisdividedintothe direc-tions parallelto thegrooveandperpendicular tothegroove [39]. Gravity supplements the shear force and has reduced stratifying effect.Consequently, convective condensation ispromoted for in-clineddownwardflow.

The transition happensathigher vapor quality forlarge mass fluxes.ThistrendofthetransitionlineisdifferentfromFig.1.The criterionofPHEsisbasedonthewettingcharacteristics,whilethe criteria oftubes are mainly derived fromjG. The possible reason

is the difference in channel geometries. The velocity distribution is not uniform within the flow section. The velocity is larger at thecenterandbecomessmallerclosetothewall.Thevaporhasa largervelocitythantheliquid.Thevaporwithlargeinertiapushes theliquidradially.Incirculartubes,theforce isnearly uniformin circumferential direction, and liquid tends to reside on the wall uniformly. Largervapor massfluxesenhance the inertia effectso astoovercomethestratifyingeffect.PHEshaveirregularflow sec-tion,andtheratioofwidthtolengthissmall.Theliquidispushed by the vapor in all directions. The corrugated surfacebreaks up theliquidfilmanddistributesthebrokendropletsrandomly. Con-sequently,acertainamountofliquidisrequiredtocompletelywet the wall. Largerliquid mass fluxesalso promote thewaviness of thefilmflowandchangetheflowpatterns.

In PHEs, the transition mass fluxes of other refrigerants are smaller than for NH3. According to Longo et al. [4], the tran-sition mass fluxes are about 20 kg m−2s −1 for HFCs and HFOs (R134a, R410A, R236fa, R1234yf, R1234ze(E)), and are around 15 kg m−2 s−1 for hydrocarbons (R600a, R290, R1270). Mancin et al. [40] stated that the transition values are around 20 kg m−2 s−1 for R410A and R407C. Thonon and Bontemps

[41]identified the transitionmassfluxesas5–13 kgm−2 s−1 for R601,R600andR290.Theseexperimentswereconductedatsmall mass fluxes, andthe HTCs ofgravity-controlled condensation are larger than convective condensation. Zhang etal. [42] concluded thatthetransitionmassfluxesareabout20kgm−2 s−1forR134a and R1234ze(E). Sarraf etal. [43] reported condensation ofR601 in the range of 9–30kg m−2 s−1. Gradual transition happens at 10–20 kg m−2 s−1 and relies on the vapor qualities. The differ-encecanbeattributedtothesurfacetension.NH3 haslarger sur-facetensionandreducesthewettability.Themassfluxneedstobe larger forcompletewetting.Consequently, gravity-controlled con-densationextendstolargermassfluxes.

4.2.Combinedcondensation

DuringthevisualizationexperimentsofNH3 condensationina PHE[10],theflowdirectionofthefluidisobservedto bea com-binationofcrossingflow andwavylongitudinalflow. Theflow in groovesisinclineddownward.As theliquidmassfluxesdecrease, a part of the wall surface is not wetted. The flow pattern be-comespartialfilm,andsome vaporisincontactwiththewall di-rectly. Partial film flow is similar to stratifiedflow intubes. The condensation deviates from convective condensation and shows similarity to gravity-controlled condensation. Convective conden-sation occursin the wetted area. In the dryzones, the conden-sationis similar to the Nusselt’s theory except for the influence of shear force. The vapor phase exerts shear force on the thin condensatefilm andenhancesthe heat transfer [44].In Eq.(15), the heat transfer correlation combines convective condensation,

α

cc,withgravity-controlledcondensation,

α

gc.



isthefractionof

convectivecondensation. Becausetheoverall heattransfer area is onlycontributedbythesetwomechanisms,thefractionof gravity-controlledcondensationcanbedeterminedas1₋



.

α

c=



α

cc+

(

1−

)

α

gc (15)

Theflow patternchangesgradually fromfull filmflow to par-tialfilmflow,andtheHTCofconvectivecondensationisidentified to be the same as givenby Eq.(11).



needs to be determined beforecalculatingthe gravity-controlledcondensation.During the condensation in horizontaltubes, forstratified flow, Dobson and Chato[14]andThomeetal.[16]estimatedthefractionof gravity-controlledcondensationbyusingthewettedangle.Cavallinietal.

[13]usedjG toquantifythefractionofgravity-controlled

conden-sation.InPHEs,thewettedangleisdifficulttomeasure.Asshown inEq.(16),



isdeterminedtobetheratioofliquidWeber num-bertothetransitionliquidWebernumber.Thefractionofthe wet-tedarea increaseswiththeliquidmassflux[45],andEq.(15) ap-proachesconvectivecondensation.



= WeL/WeL,T (16)

InEq.(17),theNusseltcorrelationiskeptasthestartingpoint

[46].The originalNusseltcorrelation isbasedon several assump-tions.Thedeviationsfromtheassumptionshaveminorinfluences, or the resulting overprediction and underprediction partly offset each other. Consequently, the correlation is applicable in wide ranges of conditions [47]. Nevertheless, most of the complicat-ing factors in the confined channels enhance heat transfer, and thus the influences need to be identified. The film thickness re-duceswhen thevaporshaves theinterface.The vaporflow accel-erates the condensate film and generates waves, contributing to film convection. PrL indicates the relative importance of

convec-tionandconduction.Co isincludedtointerprettheenhancement contributedbytwo-phase slip, andtheconstantsareobtainedby fittingthedata.

α

gc=0.36Co−0.28



g

ρ

L

(

ρ

L−

ρ

G

)

hLG

λ

3L

μ

L



Tdh



0.25 Pr0.333 L (17)

4.3.Assessmentofheattransfermodel

Fig.9comparestheproposedheattransfermodelwiththe ex-perimental data from Tao et al. [10]. 96.3% of the experimental data are predicted within ±20%. The MAE is 7.4%. The data are composedofconvectivecondensationandcombinedcondensation. Thismodel predicts accuratelythe noticeablesensitivity tovapor qualitiesandmoderatesensitivitytomassfluxes.

Tao and Infante Ferreira [6] developed an extensive experi-mentaldatabaseofcondensationinPHEs.The databaseismostly composed of HFCs, hydrocarbons and HFOs, which usually have

Table. 1

Two-phase fluid properties of typical refrigerants at T sat = 20 ◦C a

P sat P sat /P cr ρG ρL μG μL h LG σ λL Pr L Units kPa - kg •m −3 _{kg •m} −3 _{μPa •s μPa •s kJ •kg} −1 _{mN •m} −1 _{W •m} −1 _K −1 _- NH 3 857 0.075 6.70 610 9.69 134 1186 21.7 0.481 1.32 R601 57 0.017 1.72 626 6.61 230 370 16.0 0.114 4.64 R600a 302 0.083 7.91 557 7.36 159 334 10.6 0.091 4.20 R600 208 0.055 5.31 579 7.21 167 367 12.5 0.107 3.78 R290 836 0.197 18.08 500 8.01 102 344 7.6 0.096 2.84 R134a 572 0.141 27.78 1225 11.49 207 182 8.7 0.083 3.50 R410A 1447 0.295 56.80 1083 13.33 126 194 6.0 0.092 2.27 R1234ze(E) 427 0.118 22.61 1179 11.92 203 171 9.6 0.076 3.66 Air/water 857 b _{- 10.22 999 18.32 1001 - 72.8} c _{0.598 7.00}

a The fluid properties are calculated using Refprop _[55]

b This is not the saturated pressure, but the pressure used to determine the fluid properties of mixture c The surface tension of mixture is calculated using Vargaftik et al. _[56]

Fig. 9. Comparison of NH 3 condensation HTCs with the proposed model. Mean er- ror (ME): 1

n  n

1

αpre−αexp

αexp ; Mean absolute error (MAE):

1 n

 n 1

|αpre−αexp|

αexp ; Root mean

squared error (RMSE): 1 n  n 1(α pre−αexp αexp ) 2

; Per ± 20% : Percentage of experimental data within ± 20%; Per ± 30% : Percentage of experimental data within ± 30%; Per ± 50% : Percentage of experimental data within ± 50%.

muchsmallertwo-phasedensityratiothanNH3.Table1givesthe fluidpropertiesoftypicalrefrigerantsandair/watermixture.R601, R600aandR600alsohavelarge two-phase densityratio,andare usedtovalidatetheproposedheattransfercorrelations.

Sarraf et al. [43] measured the local HTCs during R601 con-densation in a PHE, andthe experimental data are presented in

Fig.10(a).HTCsincreasesharplywithvaporqualities,andareless sensitive to mass fluxes. Larger mass fluxes slightly reduce the HTCs atlow vapor qualities, andhave greater influences at high vaporqualities.Themodelpredicts wellthetrendsofvapor qual-ities.Becauseofthelargetwo-phasedensityratio,aslightriseof thevaporqualitiesgreatlyintensifiestheshearforceandenhances theheattransfer.Accordingtotheproposedmodel,theshearforce isweakatlowvaporqualities.Largermassfluxespromotefullfilm flow.Increasedwettedareasreinforcetheliquidfilm,whichserves asheattransferresistance.Thescatterislargerathighvapor qual-ities.Themodelisderivedfromdatalimitedtolowand intermedi-atevaporqualities.Moreover,themeasurementathighvapor qual-itieshaslargeuncertainties. LargerHTCsreduce theheattransfer differenceandenlargetherelativeuncertaintyforgivensensor ac-curacy. Fortunately,becauseof the smallheat transfer resistance, theheat transfer areas corresponding to high vapor qualities ac-countforasmallportionofPHEs,whichbringsaboutlimited de-signuncertainty.

Longo[48]andThononandBontemps[41]measuredthe over-all HTCs during the complete condensation of R600a, R600 and R601. Fig.10(b)and(c)comparetheexperimental datawith pre-diction. The averaged vapor quality isthe integrated meanvalue ofthewholecondenser.Shah[24]arguedthattheaveragedvapor qualityiscloserto0.4than0.5.Largermassfluxeshaveminor in-fluencesonHTCs,whichisthecompensatingeffectoflargershear force andthickerliquidfilm.Themodelismoreaccurate atlarge massfluxes,butoverpredictstheexperimentaldataatsmallmass fluxes.

In summary, the model can be extended from NH3 to other refrigerants oflarge two-phase densityratio.The model is appli-cableto the vapor qualities of0–0.8 andthe massfluxes of20– 80kg m−2 s−1.Application to highervapor qualitiesandsmaller massfluxesshouldbewithconcern.TheforemostPHEgeometries arehydraulic diametersandchevronangles,whichgenerallyspan therangeof3–8mmand25°–70°,respectively[6].Theheat trans-fermodelisatwo-phasemultiplierapproach.Theinvolved single-phase correlation identifies geometric parameters [1]. The model isexpectedtoapplytomostcommercialchevronPHEswith stan-dardgeometries.Theinfluenceofsaturatedpressureismainly at-tributed to vapor density, liquid thermal conductivity and latent heat.Themodelissuitableforlowreducedpressure.

Fig. 11showsa sensitivity analysisaccording to theproposed heat transfermodel,indicating a sharpincrease withvapor qual-ities.InFig.11(a),thecondensationmechanismiscombined con-densation at20 kgm−2 s−1 sincethe wall cannot be completely wetted. At larger massfluxes, convective condensationapplies at lowvaporqualitiesandchangesintocombinedcondensationwith increasing vapor qualities. Combined condensation takes place whenpartofthewallbecomesdry.Thetransitionvaporqualityis higherforlargermassfluxes. Whenthecondensationmechanism remains the same,HTCs increase with massfluxes. It applies for both combined condensationat highvapor qualities and convec-tivecondensationatlowvaporqualities.Forlowervaporqualities, the HTCsat 20kg m−2 s−1 are larger because theearlier transi-tion tocombined condensation.Fig. 11(b) showsthe influenceof saturatedpressures.TheHTCsareslightlylargerforlower conden-sationpressures.The largertwo-phase densityratioandviscosity differenceintensifytheshearforce.Additionally,thethermal con-ductivity of liquid is larger for low pressures. The condensation mechanismchangesclosetothevaporqualityof0.4. Lower pres-suresslightlyreducethetransitionvaporquality.

5. Developmentoffrictionalpressuredropmodel

Thefrictionalpressuredropislessaffectedbythetransitionof condensation mechanisms. Thus a unified model is developed in thisSection.The original Lockhart–Martinellimodelisfirstly pre-sented, andthen is modified to identify the influence ofsurface

Fig. 10. Validation of the new heat transfer model, Eqs. (11) and (15)–(17) , with the experimental data of (a) Sarraf et al. [43] , (b) Longo [48] , (c) Thonon and Bontemps [41] .

tensionandthecontribution ofthe vaporpressure drop.The ex-perimentaldatareportedinTaoetal.[10]areusedasthebasisfor thismodifiedapproach.

5.1.Separatedflowmodel

Both full film flow and partial film flow are separated two-phase flows. The vapor phase has larger velocity than the liquid phase. The two-phase frictional pressure drop is the sum ofthe liquidpressure drop,vapor pressure drop and interface pressure drop.Lockhart–Martinellimodelisselectedasthestartingpointto predictthefrictionalpressuredrop.

TheoriginalLockhart–Martinellimodelwasdevelopedfortubes andhastheformofEqs.(18)–(24)[20,19].Xistheratioofthe liq-uidtovaporpressuredrops.Xdeterminesthecontributionsof liq-uidandvaporpressuredrops.Thevalue ofXdependsonwhether theliquidandvaporarelaminarorturbulent.Eqs.(21)–(22)apply toturbulent-turbulentflowandlaminar-laminarflowbasedonthe single-phasepressuredropcorrelationsintubes[49].Xisa func-tionof vapor qualitiesandtwo-phase fluid properties. Theratios of densities and viscosities determine the two-phase slip at the interface. Eq.(18) can be converted into Eq.(25) by substituting

Eqs.(19)–(20).

φ

2 L =1+ C X+ 1 X2 (18)

φ

2 L =



PTP



PL (19) X2=



PL



PG = fL fG



_{1− x} x



2

_ρ

G

ρ

L (20) Xtt=



_{1− x} x



0.875

ρ

G

ρ

L



0.5

 μ

L

μ

G



0.125 (21) Xll=



_{1− x} x



0.5

ρ

G

ρ

L



0.5

 μ

L

μ

G



0.5 (22)



PL= fL G2 L 2

ρ

L Lp dh =fL G2

₍

_{1− x}

₎

2 2

ρ

L Lp dh (23)



PG=fG G2 G 2

ρ

G Lp dh =fG G2_x2 2

ρ

G Lp dh (24)



PTP=

_



PL

liqudpressuredrop

+ C

_





_

PL



PG



interfacepressuredrop

+

_



PG

vaporpressuredrop

(25)

In PHEs, the ratio of f cannot be easily represented by vapor qualityandfluidpropertiesliketubesinEqs.(21)–(22).Asshown in Eqs. (8)–(10), fL and fG are calculated based on single-phase

flow correlations. It is recommended to calculate X accordingto

Eq. (20) or calculate



PTP directly using Eq. (25). The interface

pressure drop is proportional to the geometric mean of the liq-uid and vapor pressure drops. The two-phase frictional pressure dropapproachestheliquidorvaporpressuredropwhenxis0or 1,respectively. The interface pressure drop vanishes as the fluid becomessinglephase.

During two-phase flow, theLockhartand Martinellimodel as-sumesthatthevaporflowmechanismstaysthesameasfor single-phase flow [20]. As shown in Fig. 2, the vapor pressure drop is predicted to be larger than the prediction of other models. Müller-SteinhagenandHeck [22] proposed themodel introduced inEq.(26),where



PLO and



PGO are theliquid onlyandvapor

onlypressuredrops,respectively.Thismodelislimitedbythe liq-uidandvaporpressuredropswhenthefluidbecomessinglephase.

InordertobecomparedwithLockhart-Martinellimodel,itis con-vertedintoEq.(27).Forfluidswithlarge two-phasedensityratio, thevaporpressure dropcontributesprimarily totheoverall pres-suredrop.TheLockhart-Martinelli modelinvolvesthevapor pres-suredropdirectly,whilethemodelofMüller-SteinhagenandHeck reducesthecontributionbymultiplyingwith(fGO/fG)x.InPHEs,the

single-phasefrictionalpressuredropidentifiestheinfluenceof cor-rugatedflowpassagesandhaslargevaluesbecauseofthe geome-tryinducedmomentumdissipation[1].Thedirectinclusionof va-porpressuredropover-predictsthetwo-phasepressuredrop.The proposedmodelispresentedinEqs.(28)–(29).Thecontributionof thevaporpressuredropismodifiedandisproportionaltothe va-porquality.



PTP=

(

1− x

)

1/3



PLO+2x

(

1− x

)

1/3

(

PGO−



PLO

)

+x3



PGO (26)



PTP= fLO fL

(

1− x

)

−5/3

_

_P L







liqudpressuredrop

+2x

(

1− x

)

1/3

_(

_P

GO−



PLO

)







interfacepressuredrop

+ fGO fG

x



PG







vaporpressuredrop

(27)

The original Lockhart–Martinelli model wasreported to over-predict thefrictional pressuredropwhen thesurfacetension be-comesdominant[50,51].Largersurfacetensionreducesthewetted area andflowresistance contributedby thelarge viscosityofthe liquidphase[52].Surface tensiontendsto smooththe two-phase interface and reduce the pressure drop [51]. The friction at the two-phase interface dependsonflow patterns,andtheLockhart– Martinellimodelshould be modifiedaccordingly[53].Lower val-ues of C are recommended when surface tension dominates at smalldiameter channels.In Eqs. (28)–(29),C is selectedto be 2.

Eq. (29) is obtained by substituting Eqs. (19)–(20) into Eq. (28). Thesingle-phasepressuredropiscalculatedusingthefriction fac-torobtainedfromEqs.(8)–(10).Thetwo-phase frictionalpressure dropis limitedby single-phasepressure dropwhenthe flow be-comesliquidorvapor.

φ

2 L =1+ 2 X + x X2 (28)



PTP=

_



PL

liqudpressuredrop

+ 2





PL



P_G







interfacepressuredrop

+ x

_



_

PG



vaporpressuredrop

(29)

5.2. Assessmentoffrictionalpressuredropmodel

InFig.12,theproposedfrictionalpressuredropmodelpredicts 73.8%oftheexperimentaldatawithin±20%.TheMAEis14.6%.The data are divided intofull film flow andpartial film flow accord-ingtoWeL,T=0.12,whichcorrespondtoconvectivecondensation

andcombinedcondensation,respectively.Theexperimentaldataof partial filmflow are slightlyover-predicted. Forpartial filmflow, parts ofthe wall are sheared by thevapor instead ofliquid,and theoverallflowresistanceisreduced.Moreover,theshearforceat thetwo-phaseinterfaceisreducedsincetheinterfaceshrinksand thevaporcontactsthewalldirectly.

InTable1,thetwo-phasefluidpropertiesofair-waterare sim-ilar tothose ofNH3 withlarge densityratioandsurfacetension. The flow characteristics are expected to be similar. According to theexperimentalresultsofTribbeandMüller-Steinhagen[54]and Winkelmann [28], the frictional pressure drop increases linearly

Fig. 12. Comparison of NH 3 frictional pressure drop with the present model. Mean error (ME): 1

n  n

1Ppre−PexpP ; Mean absolute error (MAE): exp

1 n

 n

1|Ppre−PexpP ; Root exp|

mean squared error (RMSE):

 1 n  n 1( Ppre−Pexp Pexp ) 2

; Per ± 20% : Percentage of exper- imental data within ± 20%; Per ± 30% : Percentage of experimental data within ± 30%; Per ± 50% : Percentage of experimental data within ± 50%.

withthevaporquality,whichisthemassflowratioofairto wa-ter. No maximum value exists atthe region of highvapor quali-ties asshownin Fig.2.The two-phase pressure drop isthe sum ofthreecomponents.Intubeswithsmoothsurface,theliquidand vaporpressuredropsarelimited,whiletheinterfacepressuredrop is noticeablebecauseof thesags andcrests.Themaximum pres-suredropresultsfromtheshearforce atthe two-phaseinterface. ThecorrugatedchannelsofPHEssignificantlyintensifythe single-phase pressure drop [1]. Bumpy two-phase interface is unlikely tofurtheraggravatethemomentumdissipationsuperposedtothe wallfriction.Theinterfacepressuredropbecomessecondary.Thus the two-phase pressure drop is proportional to vapor qualities. Thesedatacannotbeusedtovalidatetheproposedmodelbecause theoperatingconditionsarenotreported[54,28].

The model is limited to two-phase NH3 flow. It is expected to cover the vapor qualities of 0–1 and the mass fluxes of 20– 80kg m−2 s−1.Experimental dataathighvaporqualitiesare not obtained.Butthemodelapproachesvaporonlypressuredropwith

increasing vapor qualities,which agrees withphysical interpreta-tion. Similar to the proposed heat transfer model, the pressure dropmodelappearstobe applicabletomostcommercialchevron PHEsand low reducedpressure. The sensitivityto geometric pa-rametersisincludedinthesingle-phasecorrelations[1].

Fig.13 presents thesensitivity offrictional pressure drop cal-culatedfromtheproposed model.Thefrictionalpressuredrop in-creasesdramaticallywithvaporqualitiesasthevaporphase occu-piesalargerportionoftheflowsection.InFig.13(a),largermass fluxesincrease the frictionalpressure drop significantly and con-tributetoalatertransitionofflowpatternsintermsofvapor qual-ities. As shownin Fig. 13(b),the saturated pressure hasa minor influenceatlowvaporqualitieswherethefrictionalpressuredrop ismainly contributedby theliquidphase. Higher saturated pres-surehaslarger vapordensityandreducesthevolumeflux signif-icantly athighvapor qualities. Thus the slipand shearforce be-tweenphasesdecrease.

6. Conclusions

Thispaper analyzes the wetting characteristics andinterfacial propertiesoftwo-phaseflowpatternsduringNH3condensationin PHEs.Based onthe analysis, heat transfer andfrictional pressure dropmodels arederivedfromtheexperimentaldatapresentedin apreviouspaperoftheauthors[10].

• TheheattransfermodelispresentedinEqs.(11)and(15)–(17). Itdistinguishesconvectivecondensationandcombined conden-sation. Convectivecondensationhappensforfullfilmflow,and a two-phase multiplier correlation has been developed. Com-bined condensation takesplace forpartial film flow. Theheat transfer is composed of convective condensation and gravity-controlled condensation. The HTCs of gravity-controlled con-densation deviate from Nusselt’s theory because of the two-phaseshearforceandliquidconvection.Thetransitioncriterion ofcondensationmechanismsisWe_L,T=0.12.Acrossvalidation showsthatthemodelisapplicableforotherrefrigerantsof sim-ilarfluidpropertiestoNH3.

• A unified model of separated flow is developed for frictional pressure drop,whichisshowninEqs.(28)–(29). TheLockhart and Martinelli modelis modified to identify the reduction of vaporpressuredropandtheinfluenceofsurfacetensiononthe interfacepressuredrop.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

XuanTao:Conceptualization,Methodology,Formalanalysis, In-vestigation, Data curation, Visualization, Writing - original draft.

CarlosA.Infante Ferreira: Conceptualization, Resources, Project administration,Fundingacquisition,Methodology,Writing-review &editing,Supervision.

Acknowledgments

ThisprojecthasbeendevelopedincooperationwithAllseas En-gineeringB.V.Theauthorsacknowledgethefinancialsupportfrom theChina ScholarshipCouncil and fromthe KoudeGroep Delft /

Wageningen.

References

[1] H. Martin , Pressure drop and heat transfer in plate heat exchangers, in: P. Stephan, H. Martin, S. Kabelac, D. Mewes, M. Kind, K. Schaber (Eds.), VDI Heat Atlas, Springer, Dusseldorf, 2010, pp. 1516–1521. 2nd Edition .

[2] X. Tao , M.P. Nuijten , C.A Infante Ferreira , Two-phase vertical downward flow in plate heat exchangers: flow patterns and condensation mechanisms, Int. J. Refrig. 85 (2018) 489–510 .

[3] S. Buscher , Visualization and modelling of flow pattern transitions in a cross– corrugated plate heat exchanger channel with uniform two-phase distribution, Int. J. Heat Mass Transf. 144 (2019) 118643 .

[4] G.A. Longo , G. Righetti , C. Zilio , A new computational procedure for refrigerant condensation inside herringbone-type brazed plate heat exchanger, Int. J. Heat Mass Transf. 82 (2015) 530–536 .

[5] F. Táboas , M. Vallès , M. Bourouis , A. Coronas , Flow boiling heat transfer of ammonia/water mixture in a plate heat exchanger, Int. J. Refrig. 33 (2010) 695–705 .

[6] X. Tao , C.A Infante Ferreira , Heat transfer and frictional pressure drop during condensation in plate heat exchangers: assessment of correlations and a new method, Int. J. Heat Mass Transf. 135 (2019) 996–1012 .

[7] W.S. Kuo , Y.M. Lie , Y.Y. Hsieh , T.F. Lin , Condensation heat transfer and pressure drop of refrigerant R-410A flow in a vertical plate heat exchanger, Int. J. Heat Mass Transf. 48 (25) (2005) 5205–5220 .

[8] H. Kumar , in: The Design of Plate Heat Exchangers for Refrigerants, Proceed- ings of Institute of Refrigeration., London, The UK, 1992, pp. 50–55 . [9] C.B. Panchal , T.J. Rabas , Thermal performance of advanced heat exchangers for

ammonia refrigeration systems, Heat Transf. Eng. 14 (1993) 42–57 .

[10] X. Tao , E. Dahlgren , M. Leichsenring , C.A Infante Ferreira , NH3 condensation in a plate heat exchanger: Experimental investigation on flow patterns, heat transfer and frictional pressure drop, Int. J. Heat Mass Transf. 151 (2020) 119374 .

[11] B.M. Fronk , S. Garimella , Condensation of ammonia and high-tempera- ture-glide ammonia/water zeotropic mixtures in minichannels–part I: mea- surements, Int. J. Heat Mass Transf. 101 (2016) 1343–1356 .

[12] C.Y. Park , P. Hrnjak , NH3 in-tube condensation heat transfer and pressure drop in a smooth tube, Int. J. Refrigeration. 31 (2008) 643–651 .

[13] A. Cavallini , D.D. Col , L. Doretti , M. Matkovic , L. Rossetto , C Zilio , G Censi , Con- densation in horizontal smooth tubes: a new heat transfer model for heat ex- changer design, Heat Transfer Eng. 27 (2006) 31–38 .

[14] M.K. Dobson , J.C. Chato , Condensation in smooth horizontal tubes, ASME J. Heat Transf. 120 (1998) 193–213 .

[15] J.E. Hajal , J.R. Thome , A. Cavallini , Condensation in horizontal tubes, part 1: two-phase flow pattern map, Int. J. Heat Mass Transf. 46 (2003) 3349–3363 . [16] J.R. Thome , J. El Hajal , A. Cavallini , Condensation in horizontal tubes, part 2:

new heat transfer model based on flow regimes, Int. J. Heat Mass Transf. 46 (2003) 3365–3387 .

[17] M.M. Shah , An improved and extended general correlation for heat transfer during condensation in plain tubes, HVAC&R Res. 15 (2009) 889–913 . [18] B.M. Fronk , S. Garimella , Condensation of ammonia and high-tempera-

ture-glide ammonia/water zeotropic mixtures in minichannels–part II: heat transfer models, Int. J. Heat Mass Transf. 101 (2016) 1357–1373 .

[19] R.W. Lockhart , R.C. Martinelli , Proposed correlation of data for isothermal two-phase, two-component flow in pipes, Chem. Eng. Prog. 45 (194 9) 39–4 8 . [20] D. Chisholm , A theoretical basis for the Lockhart–Martinelli correlation for

two-phase flow, Int. J. Heat Mass Transf. 10 (1967) 1767–1778 .

[21] L. Friedel , Improved friction pressure drop correlations for horizontal and ver- tical two-phase flow, The European Two-Phase Flow Group Meeting, Ispra, Italy, 1979 .

[22] H. Müller-Steinhagen , K. Heck , A simple friction pressure drop correlation for two-phase flow in pipes, Che. Eng. Process. 20 (1986) 297–308 .

[23] B.M. Fronk , S. Garimella , Heat transfer and pressure drop during condensation of ammonia in microchannels, in: the 3rd ASME International Conference on Micro/Nanoscale Heat and Mass Transfer, Atlanta, America, 2012 .

[24] M.M. Shah , A general correlation for heat transfer during film condensation inside pipes, Int. J. Heat Mass Transf. 22 (1979) 547–556 .

[25] K.W. Moser , R.L. Webb , B. Na , A new equivalent Reynolds number model for condensation in smooth tubes, ASME J. Heat Transf. 120 (1998) 410–417 . [26] K. Nilpueng , S. Wongwises , Two-phase gas-liquid flow characteristics inside a

plate heat exchanger, Exp. Therm. Fluid Sci. 34 (2010) 1217–1229 .

[27] C. Tribbe , H.M. Müller-Steinhagen , Gas/liquid flow in plate-and-frame heat ex- changers - part II: two-phase multiplier and flow pattern analysis, Heat Transf. Eng. 22 (2001) 12–21 .

[28] D. Winkelmann , Condensation of pure refrigerants and their zeotropic mix- tures in plate heat exchangers. Ph.D. thesis, Commissiariat à l’energie atomique (CEA) (2010) .

[29] V.G. Nino , P.S. Hrnjak , T.A. Newell , Characterization of two-phase flow in mi- crochannels, ACRC Report TR202, Air Conditioning and Refrigeration Center, University of Illinois, Urbana-Champaign, 2002 at .

[30] D.C. Adams , P.S. Hrnjak , T.A. Newell , Pressure drop and void fraction in mi- crochannels using carbon dioxide, ammonia, and R245fa as refrigerants, ACRC Report TR221, Air Conditioning and Refrigeration Center, University of Illinois, Urbana-Champaign, 2003 at .

[31] X. Tao , J.A. Kirkenier , C.A Infante Ferreira , Condensation of NH3 within a plate heat exchanger of small diameter channel, the 6th ASME International Confer- ence on Micro/Nanoscale Heat and Mass Transfer, 2019 paper 3920 . [32] E.W. Jassim , T.A. Newell , J.C. Chato , Refrigerant pressure drop in chevron and

bumpy style flat plate heat exchangers, Exp. Therm. Fluid Sci. 30 (2006) 213–222 .

[33] S.M. Zivi , Estimation of steady-state steam void-fraction by means of the prin- ciple of minimum entropy production, ASME J. Heat Transf. 86 (1964) 247–251 . [34] S.Z. Rouhani , E. Axelsson , Calculation of void volume fraction in the subcooled

and quality boiling regions, Int. J. Heat Mass Transf. 13 (1970) 383–393 . [35] M.J. Wilson , J.C. Chato , T.A. Newell , A.G. Kireta , A Study of Refrigerant Pressure

Drop and Void Fraction in Flattened Copper Tubes, International Refrigeration and Air Conditioning Conference., Purdue, America, 20 0 0 .

[36] K.A. Triplett , S.M. Ghiaasiaan , S.I. Abdel-Khalik , A. LeMouel , B.N. McCord , Gas–liquid two-phase flow in microchannels part II: void fraction and pres- sure drop, Int. J. Multiph. Flow. 25 (1999) 395–410 .

[37] X. Tao , E. Dahlgren , C.A Infante Ferreira , Condensation Heat Transfer and Pres- sure Drop of NH3 And NH3/H2O Within a Plate Heat Exchanger, 25th IIR In- ternational Congress of Refrigeration, Montreal, Canada, 2019 paper 727 . [38] J. Hart , P.J. Hamersma , J.M. Fortuin , Correlations predicting frictional pressure

drop and liquid holdup during horizontal gas-liquid pipe flow with a small liquid holdup, Int. J. Multiph. Flow. 15 (1989) 947–964 .

[39] B.X. Wang , X.Z. Du , Study on laminar film-wise condensation for vapor flow in an inclined small/mini-diameter tube, Int. J. Heat Mass Transf. 43 (20 0 0) 1859–1868 .

[40] S. Mancin , D. Del Col , L. Rossetto , Condensation of superheated vapour of R410A and R407C inside plate heat exchangers: experimental results and sim- ulation procedure, Int. J. Refrig. 35 (2012) 2003–2013 .

[41] B. Thonon , A. Bontemps , Condensation of pure and mixture of hydrocarbons in a compact heat exchanger: experiments and modelling, Heat Transf. Eng. 23 (2002) 3–17 .

[42] J. Zhang , M.R. Kærn , T. Ommen , B. Elmegaard , H. Fredrik , Condensation heat transfer and pressure drop characteristics of R134a, R1234ze (E), R245fa and R1233zd (E) in a plate heat exchanger, Int. J. Heat Mass Transf. 128 (2019) 136–149 .

[43] K. Sarraf , S. Launay , G El Achkar , L Tadrist , Local vs global heat transfer and flow analysis of hydrocarbon complete condensation in plate heat exchanger based on infrared thermography, Int. J. Heat Mass Transf. 90 (2015) 878–893 . [44] M.K. Dobson , Heat transfer and flow regimes during condensation in horizon-

tal tubes, ACRC Report TR57, Air Conditioning and Refrigeration Center, Univer- sity of Illinois, Urbana-Champaign, 1994 at .

[45] T. Ahn , J. Moon , B. Bae , J. Jeong , B. Bae , B. Yun , An empirical model of the wetted wall fraction in separated flows of horizontal and inclined pipes, Chem. Eng. Sci. 178 (2018) 260–272 .

[46] W. Nusselt , Die oberflachenkondensation des wasserdampfes, Z. Ver. D. Ing. 60 (1916) 541–546 .

[47] J.W. Rose , Fundamentals of condensation heat transfer: laminar film conden- sation, JSME Int. J. Ser. 2 31 (1988) 357–375 .

[48] G.A. Longo , Heat transfer and pressure drop during hydrocarbon refrigerant condensation inside a brazed plate heat exchanger, Int. J. Refrig. 33 (2010) 944–953 .

[49] W. Kast , E.S. Gaddis , K. Wirth , J. Stichlmair , Pressure drop in single phase flow, in: P. Stephan, H. Martin, S. Kabelac, D. Mewes, M. Kind, K. Schaber (Eds.), VDI Heat Atlas, Springer, Dusseldorf, 2010, pp. 1055–1115. 2nd Edition .

[50] H.J. Lee , S.Y Lee , Pressure drop correlations for two-phase flow within hori- zontal rectangular channels with small heights, Int. J. Multiph. Flow. 27 (2001) 783–796 .

[51] E. Ungar , J. Cornwell , Two-phase pressure drop of ammonia in small diam- eter horizontal tubes, in: AIAA 17th Aerospace Ground Testing Conference, Nashville, America, 1992 .

[52] Y. Chen , K.S. Yang , Y.J. Chang , C.C. Wang , Two-phase pressure drop of air–wa- ter and R-410A in small horizontal tubes, Int. J. Multiph. Flow. 27 (2001) 1293–1299 .

[53] Y.S. Muzychka , M.M. Awad , Asymptotic generalizations of the Lockhart— Martinelli method for two phase flows, ASME J. Fluid. Eng. 132 (2010) 031302 . [54] C. Tribbe , H.M. Müller-Steinhagen , Gas/liquid flow in plate-and-frame heat ex- changers - part I: pressure drop measurements, Heat Transf. Eng. 22 (2001) 5–11 .

[55] Lemmon E.W., Huber M.L., and McLinden M.O. (2013). NIST standard refer- ence database 23: reference fluid thermodynamic and transport properties- REFPROP, version 9.1, national institute of standards and technology, standard reference data program, Gaithersburg.

[56] N.B. Vargaftik , B.N. Volkov , L.D. Voljak , International tables of the interfacial tension of water, J. Phys. Chem. Ref. Data. 12 (1983) 817–820 .