Modelling of Multistream LNG Heat

7/28/2019 Modelling of Multistream LNG Heat

1/67

Natural Gas TechnologyJune 2011

Kjell Kolsaker, EPT

Submission date:

Supervisor:

Norwegian University of Science and Technology

Department of Energy and Process Engineering

Modelling of Multistream LNG HeatExchangers

Joan Soler Fossas


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I

AbstractThe main goal of this thesis is to find out if a liquefied natural gas multistream heat

exchanger

numerical

model

is

achievable.

This

should

include

several

features

usually

neglected in nowadays available heat exchanger models, such as flow maldistribution,

changes in fluidpropertiesandheatexchangerdynamicbehaviour. Inorder toaccomplish

thatobjectiveasimplercaseismodelled.Effortsareputinachievingnumericalstability.

A counter flow natural gas and mixed refrigerant heat exchanger is modelled. Some

importantcharacteristicsoftheobtainedmodelare:(1)itallowsadynamicstudyoftheheat

exchanger, (2)mass flowrate isaconsequenceof inletandoutletpressuredifference, (3)

fluidpropertieschangeistakenintoaccount,(4)itpresentsatimestepcontrolfunctionand

(5)fluidmovementisnotneglected.

Someinterestingnumericalbehavioursincludedinheatexchangersmodelsdesignthathave

beenobservedduringthecourseofthisthesisarediscussed.Forinstance,thecomparisonof

theeffectsofchoosingoneheattransferscorrelationoranother.

Dynamic response of themodelled heat exchanger during start up and during an abrupt

changeinmixedrefrigerantinlettemperatureareshownanddiscussed.


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II

PrefaceThisMasterthesiswascompletedattheDepartmentofEnergyandProcessEngineeringof

theNorwegianUniversityof Science andTechnology (NTNU) from February2011 to June

2011.

First, Iwould like to thankmy supervisor,NTNUprofessorKjellKolsaker, forhishelpand

guidelinesduringthecourseofthisMasterthesis. Ispeciallyappreciatethathehasalways

beenopenedtomyquestionsanddoubts,nomatterthetimeittooktosolvethem.

Iwouldalso like to thank toivindWilhelmsen,whobelongs toSINTEF staff.Hisadvices

duringallthesemesterhelpedmea lot. Specially,attheend,whenhishelp instructuring

thethesisbecamereallyvaluable.

Iwouldliketothankthiscountryanditspeople,NorwayandtheNorwegians,forgivingan

allday

example

of

the

well

done

work.

Iwouldalso like to thankmy friends, inNorwayandabroad,becauseno thesiscouldgive

whatafriendshipdoes.

Iwouldliketothankmyparentsfortheexcellentexamplethattheyhavealwaysbeen,for

theunconditional lovethat theyhavealwaysshownand forhow fortunate theymakeme

feel.

Finally,IwouldliketothankNriabecauseshealwaysmakesthethingslookbrighter.


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III

TableofContentsAbstract.......................................................................................................................................I

Preface........................................................................................................................................II

Listof

figures

..............................................................................................................................

V

Listoftables..............................................................................................................................VI

Nomenclaturetable.................................................................................................................VII

1 Backgroundandobjective..................................................................................................1

1.1 Introduction.................................................................................................................1

1.2 Aimofthethesis..........................................................................................................3

1.3 Descriptionofthethesis..............................................................................................4

2

Preliminaryconcepts

..........................................................................................................

5

2.1 Heatexchangers..........................................................................................................5

2.1.1 Heattransferandpressuredropcharacteristics.................................................5

2.2 Finitedifferencemethods.........................................................................................11

3 Calculations.......................................................................................................................14

3.1 Caseofstudy..............................................................................................................14

3.1.1 Counterflowheatexchanger:Naturalgasliquefaction....................................15

3.2 Themodel..................................................................................................................20

3.2.1 Procedure...........................................................................................................20

3.2.2 Structureandoperation.....................................................................................21

4 Resultsanddiscussion......................................................................................................32

4.1 Evaluationofthemodelrobustnessandtimeconsumption....................................32

4.1.1 Phasechangeathighpressure...........................................................................32

4.1.2 GnielinskiandDittusandBoelterheattransfercorrelationsnumericaleffects35

4.1.3 Advantagesofthevaryingrelaxingcoefficient..................................................37

4.1.4 Effectoftimesteplengthandgridmeshingonthenumericalstability............37

4.2 Caseresults................................................................................................................39

4.2.1 Thermaldynamicsduringthestartup...............................................................41

4.2.2 Thermaldynamicsduringanabruptchangeintheshellinlettemperature.....44

5 Conclusionsandrecommendationsforfurtherwork......................................................48

5.1 Conclusions................................................................................................................48

5.2 Recommendationsforfurtherwork..........................................................................49


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IV

6 Bibliography......................................................................................................................50

7 Appendix...........................................................................................................................51


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V

ListoffiguresFigure1 TypicalBreakdownofLiquefactionPlantCapitalCost[1].........................................1

Figure2 SeveraltypicalassumptionsimpactontheHEdesigndependingontheHE

effectiveness[2].........................................................................................................................2

Figure3

Two

phase

flow

types

.................................................................................................

6

Figure4 Geometricinterpretationofthedifferentapproximations.....................................12

Figure5 Shellandtwotubesmultistreamheatexchangerscheme......................................14

Figure6 Shellandtubeheatexchangerscheme...................................................................15

Figure7 Naturalgasinputpressuredependenceontime....................................................17

Figure8 Mixedrefrigerantinlettemperaturedependenceontime.....................................17

Figure9 Mixedrefrigerantinletpressuredependenceontime...........................................18

Figure10 Discretizedschemeoftheheatexchangermodelled............................................21

Figure11 Modeloperationscheme.......................................................................................23

Figure12Tubesidecellenergybalancescheme.....................................................................25

Figure13 Numericallystablesituation..................................................................................26

Figure14 Numericallyunstablesituation..............................................................................26

Figure15 Fixedcoefficientrelaxingsolutionscheme............................................................27

Figure16Behaviouroftherelaxingthesolutionfunctiondevelopedandusedinthemodel

..................................................................................................................................................28

Figure17 Modeltimestepfunctionscheme.........................................................................30

Figure18 Vapourfractionvs.Enthalpyat50bar...................................................................32

Figure19 Heattransfercoefficientdependencevs.Enthalpyat50bar................................33


Figure21 Heattransfercoefficientvs.Enthalpyat54bar.....................................................34


Figure23 Heattransfercoefficientvs.Enthalpyat50bar.....................................................35

Figure24 NumberofiterationsnecessarytoachieveconvergencewhenGnielinski

correlationisused....................................................................................................................36

Figure25 NumberofiterationsnecessarytoachieveconvergencewhenDittusandBoelter

correlationisused....................................................................................................................36

Figure26

Number

of

iterations

necessary

to

achieve

convergence

when

afixed

relaxing

coefficientvalueisused...........................................................................................................37

Figure27 Mixedrefrigerantinlettemperaturedependenceontime...................................39

Figure28 Codeofcoloursusedwhenplottingdifferentcellsofasamestream..................39

Figure30 Mixedrefrigeranttemperatureinseveralshellcells.............................................40

Figure31 Naturalgasmassflowrateduringtime.................................................................41

Figure31 Naturalgascellstemperaturesduringt


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VI

ListoftablesTable1Naturalgasandmixedrefrigerantscomposition.....................................................15

Table2Tubeandshellgeometricaldata..............................................................................16

Table3 Aluminiumproperties...............................................................................................16


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VII

Nomenclaturetable, , Friedelscorrelationconstants

hydraulicdiameter

frictionfactor polynomiallinkingfunction Froudenumber gravitationalacceleration massflux

heattransfercoefficient

intensiveenthalpy length massflowrate totalcellsnumber pressure

heatflux

Reynoldsnumber surface time specificvolume volume

Webbernumber

thermodynamicequilibriumquality streamwisecoordinateGreeksymbols voidfraction channelaspectratio

thermalconductivity

viscosity

density

surfacetension

twophasepressuredropmultiplierSuperscripts twophase timestepSubscripts

accelerational

contraction expansion fluid saturatedvapour cellposition

channelinlet

saturatedliquid laminarflow liquidonly channeloutlet Shell

singlephase

tube total twophase transitionalflow turbulentflow

wall


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2

is the result of hundreds of thin shell and two tubes working in parallel. Usually, a low

pressurerefrigerantsuchasapureormulticomponentrefrigerantflowsdowntheshellside.

Thetubesideisoccupiedbythehotstreams;usuallyeachstreamfillsonetube.

Heat exchangers are the main components in cryogenic processes. In LNG plants they

represent2030%of the investmentcost [2,3]. Inaddition, theirperformanceaffects the

sizinganddesignofotherequipment,namelycompressorsandtheirpowerdrivers.

Ithasbeenstudiedbyseveralauthorshowthermodynamicconsiderationsmakecryogenic

processesverysensitivetotheHEperformance[2,4,5].

AcryogenicHEinaLNGplantisexpectedtohaveanefficiencyhigherthan90%.Thismeans

thatthedesignhastobereallyaccurateandithasbeennoticedthatseveraleffectsthatare

neglectedwhendesigningahightemperatureHEshouldnotbeneglectedwhendesigninga

HEfor

cryogenic

applications.

These nonnegligible effects are: changes in fluid properties, flow maldistribution,

longitudinal thermal conduction and heatinleakage. The consideration of this effects

dependontheeffectivenessthataHEisexpectedtoperform:

Figure2 SeveraltypicalassumptionsimpactontheHEdesigndependingontheHEeffectiveness[2]

Thus,asithasbeenexplainedMSHEsplayanimportantroleintheoperationandthedesign

ofaLNGplant. It is theengineerspriority to find theway to improve theirdesignand to

predicttheirperformance.Thereforelotsofheatexchangerssimulationsmodelshavebeen

developed in the last decades and are used extensively for both designing plants and

evaluatingtheactualperformanceofaplant.

However,over

the

last

years

it

has

been

seen

that

there

is

aneed

to

improve

HE

models

for

cryogenicapplications.This isspecially thecasewhen itcomes toreducing thenumberof


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3

assumptionsof idealconditions thatareused.Oneof thereasons isthepursuingofmore

energy,costandspaceefficientplants.Featuresusedtoachievethis includeamongothers

mixed refrigerant andmore compact heat exchangers. These featureswill inmany cases

lead to conditions where ideal assumptions are no longer valid. If this is not taken into

accountwhen

designing

the

plant,

it

could

give

operational

challenges

and

as

well

lead

to

an

energyconsumptionthatishigherthananticipated.

Becauseof itsownnature,MSHEswithphasechange involveseveralphysicalphenomena

thatshouldbeconsideredwhenmodellingthem.

Characteristicsitisimportanttoaccountforinthedesignphaseinclude:

Distributionoftwophasefluidsintoseveralparallelchannels. Instabilitiesinchannelswithevaporation/condensation. Maldistributionoftwophasefluidswithinheatexchangerschannels. Two phase non equilibrium conditions (both thermodynamic and fluid dynamic

effectslikeslip).

Heatexchangerdynamicbehaviour,forinstanceatstartupandshutdown.Both steady state and dynamic heat exchanger models are widely used in the design

processesofheatexchangersandoverallLNGproductionplants.Detailedplantoperability

analysisrequiresgooddynamicheatexchangermodels.Astheplantcomplexityincreases,it

isseenthatthesekindofanalysisareimportantpartsoftheplantprocessaswell.

Theheat

exchanger

models

which

are

used

are

usually

based

on

enthalpy

balance

calculations or on heat transfer and pressure drop correlations. The latter method is

normallymoredetailedandgeometrydependentthanthefirstone.

The correlation basedmodelswill in addition to correlations, based on empirical results,

contain some kind of assumptions in order to simplify the heat exchanger modelling.

Assumptions of equal two phase distribution, stable flow conditions and thermodynamic

equilibriumarecommon.

SINTEF Energy Research is currently running a competence building project (KMB: Low

emissionsLNG systems)withcontribution fromNTNUand industrypartners, focusingon

bothfundamentalphysics,heatexchangerdesignandsystemoptimisation.

This thesis aims to produce a model capable of including part of the problematic here

explained.ThegoalistoproduceatoolthatallowsimprovedLNGprocessdesigns.

1.2 AimofthethesisTheobjectiveofthisthesisistofindoutifitispossibletodevelopaMSHEnumericalmodel

that includes most of the problematic already explained. This model should simulate a

multistreamheat

exchanger

working

with

natural

gas.

The

model

should

be

able

to

simulate


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4

satisfactorily phase changes in transient conditions. Minimum simplifications as possible

shouldbeassumedinfluidpropertiesfield.

Main model characteristics thatwill be appreciated are numerical stability, accuracy and

CPU timeconsumption.Specially, firstonewillbethemost importantsincetheother two

canbeimprovedlaterifastablemodelisdeveloped.

Toachievethisconclusion,asimplermodelwillbedeveloped.Acounterflowtwostreams

shellandtubeheatexchangerwillbemodelled.Thismodelshouldbea toolthatallowsa

heatexchangerdesignerknowmoreabouttheLNGprocessdynamics.

Effortswillbeputinachievingaworkingnumericallystablemodel.Experimentalvalidityof

theresultsisnotthegoalofthisthesis,hence,therecanbesomeoftheuseddatathatmay

notbeasaccurateasdesired inafinalheatexchangerdesign.Besides,theoptimisationof

themodel

can

be

done

afterwards

in

further

work

researches.

1.3 DescriptionofthethesisInchapter2isanexplanationofthepreliminaryconceptsthatarenecessarytounderstand

thisthesis.Chapter2.1focusesontheheatexchanger,whatitis,whereitisusedandhowit

isdesigned. Sinceaheatexchangermodel isdeveloped in this thesis, chapter2.1.1 goes

deeperinthedesignissueandtheheattransferandpressuredropcorrelationsusedinthe

model are explained there. Chapter 2.2 is a basic theory explanation of finite different

methodsanditsbasicconcepts.

Calculations belong to chapter 3. First, in chapter 3.1, the case modelled is explained in

detail.Chapter3.2 isanexplanationofthemodelsolutionmethodology,theprocedureto

achievethesatisfactorycodeanditsstructureandoperationaredescribedinit.

Chapter4consistsoftheresultsandobservationsthatareproductofthisthesis.Inchapter

4.1 some relevant numeric characteristics that have been observed during the model

developmentarepresentedanddiscussed.Chapter4.2 isan illustrationanddiscussionof

theresultsobtainedfromthecaseofstudy.

Chapter5summarizes

the

most

important

and

relevant

issues

that

are

product

of

this

thesis.

Afutureworkproposalisdoneinchapter5.2.


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5

2 Preliminaryconcepts2.1 HeatexchangersShah and Sekulic give the following heat exchanger definition [6]:A heat exchanger is a

devicethat

is

used

for

transfer

of

thermal

energy

(enthalpy)

between

two

or

more

fluids,

betweena solid surfaceanda fluid,orbetween solidparticulatesanda fluid,atdiffering

temperaturesandinthermalcontact,usuallywithoutexternalheatandworkinteractions.

Inmostheatexchangers,thefluidsareseparatedbyaheattransfersurface,and ideallydo

notmix.Aheatexchangerconsistsofaheatexchangingelementssuchascoreoramatrix

containing the heat transfer surface, and fluid distribution elements such as headers,

manifolds,tanks,inletandoutletnozzles...Usually,therearenomovingparts.

Theheattransfersurfaceisasurfaceoftheheatexchangercorethatisindirectcontactwith

fluidsandthroughwhichheatistransferredbyconduction.

Thebalancedifferentialequationthatgovernsthefluidenthalpyvariationwhenthefluidis

incontactwiththeheattransfersurfaceisthefollowing:

Eq.1Wherethelasttermintherighthandoftheequation isthedifferentialheatflow,whichis

definedbythenextequation:

, Eq.2Notice that theheat transfer coefficient is then theparameterwhichwilldefine theheat

exchangervolumeinadesign,ortheonethatwillgovernitsefficiencywhenevaluatingthe

HEperformance.

AnotherimportantpartofaHEdesignisthepressuredropthatiscausedbythefluidgoing

throughit.Pressuredropwilldirectlydefinethecapitalandoperationalcostsoftheprocess

wheretheheatexchangerisused.Afluidthatsuffersahighpressuredropgoingthrougha

HEwillprovokehighcostsincompressorsorpumpsinvestmentandpowerconsumption.

Thus,heat transferandpressuredrop inheatexchangerare two important features that

which need a special attention of the designer. Therefore, chapter below gives a further

explanationofthemandofthewaytheyhavebeenmodelledinthisthesis.

2.1.1 HeattransferandpressuredropcharacteristicsHeattransferandpressuredrop insideaheatexchangeraretwocomplexphenomenathat

havebeenstudiedindecades.Researchershavebeenconcernedwithdevelopingpredictive

modelsfor

several

industries,

such

as

traditional

ones

like

steam

and

nuclear

power


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6

generation, chemical and petroleum, etc. or newer ones like electronics and all micro

channelrefrigerationsystemsthatitincludes.

2.1.1.1 TwophaseflowtypesIt

is

known

that

pressure

drop

and

heat

transfer

characteristics

depend

strongly

on

two

phase flow types.Dependingonmassandheat flowsandon vapour fraction,apart from

othervariables,twophaseflowtypesare:

Bubbleflow:Liquidfillsallthepipewhilesomesmallvapourbubblesareinamixturewithit.

Slugflow:Bigvapourbubblestravelthroughthepipeseparatedbyaliquidfilm. Annularflow:Liquidcoversonlyathinfilmthroughthepipewallwhilevapourflows

throughaninsidechannelwithsomedropletsinit.

Figure3 Twophaseflowtypes

Intheupperfigurethedifferentflowtypescanbeobserved.Churnandwispyannularflow

canbeconsideredmiddletermsbetweenslugandannularflow.

Twophase

correlations

described

below

base

their

calculations

on

the

influence

of

each

flow

typewhenaphasechangeoccurs.

2.1.1.2 Heattransfer2.1.1.2.1 Single phaseTwo popular single phase heat transfer coefficient correlations for turbulent flow ( 3000)inapipearetheonedevelopedbyDittusandBoelter[7]andGnielinski[8].DittusandBoeltercorrelationis:

0.023.. Eq.3


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AndGnielinskicorrelationis:

81000 112.78 1 Eq.4

1.82 log 1.64 Eq.5It is decided to use Gnielinski heat transfer correlation since it was noticed after some

simulations thatDittus andBoelter correlation introducedhighernumericaloscillations in

thesystem.

Hausen[9]correlationwillbethesinglephaselaminarflowheattransfercorrelationusedin

themodel.So,for

2300:

3.66 0.0668 1 0 . 0 4 Eq.6IftheReynoldsnumberiswithintheupperlimitofHausencorrelationandthelowerlimitof

theGnielinskicorrelation,Nusseltnumberwillbecalculatedasan interpolationofbothto

guaranteeNusseltcontinuity.For2300 3000:

, , Eq.7

It isreminded thatonce theNusseltnumberhasbeendefined,heat transfercoefficient is

foundbythefollowingequation:

Eq.82.1.1.2.2 Two phase2.1.1.2.2.1 Evaporation

Liu and Winterton [10] correlation is used when calculating evaporation heat

transfer.

. Eq.9Where is DittusBoelter single phase heat transfer coefficient for the liquid

(alreadyexplained),isthenucleateboilingcoefficient: 55.log... Eq.10

And

and

are:


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1 1. Eq.11

1 0. 55..

Eq.12

2.1.1.2.2.2 CondensationThere are less available condensation heat transfer correlations than evaporation

ones.Shahfilmcondensationcorrelation[11]hasbeenchosenhereforbeinganoldbroadly

testedcorrelation.

1 . 3.8.1 .. Eq.13Where

is the vapour quality,

is reduced pressure

/ and

is

DittusBoelter

single

phase

heat

transfer

coefficient

for

the

liquid.

2.1.1.3 PressuredropConsideringthestudiesthatLeeandMudawardevelopedinevaporatorsfield[12],total

pressuredropinaheatexchangercanbeexpressedas:

, ,/ Eq.14Pressure drop inside a heat exchanger where phase change is happening includes the

sudden contraction loss at the channels inlet and a sudden expansion recovery at the

channelsoutlet.However,thesetwotermsarenotincludedwhencalculatingpressuredrop

in this thesisbecause theyareconsidered tobeoutof thestudy.Asinglephasepressure

drop term is included corresponding to the flow before the phase change conditions are

achieved,meaningavapourprecoolingincondensersoraliquidpreheatinginevaporators.

The two phase pressure drop term consists of frictional and accelerational components.

Finally,another singlephase flow term is includedcorresponding to subcoolingor super

heating that can occur in condensers orevaporators, respectively,when phase change is

completed.

2.1.1.3.1 Twophase pressure drop models2.1.1.3.1.1 HomogeneousEquilibriumModel(HEM)Thehomogenousequilibriummodelisbasedontheassumptionthatthetwophasemixture

behavesasapseudosinglephase fluidwithmeanpropertiesweighted relative tovapour

and liquid content, and that only latent heat may be exchanged between the phases.

Propertyvariationsresultingfrompressurechangesalongthechannelresultincomplicated

terms that account kinetic energy, flashing and compressibility. The resulting pressure

gradientmaybeexpressedas:


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2 4 1

1 1 1 1 1

1

1 Eq.15

The first term in the numerator of Eq. (13) is the frictional gradient and the second the

accelerational.Thedenominatorincludeskineticenergy,flashingandcompressibilityterms.

The twophasepressuredropcanbedeterminedby integratingEq. (13)numericallyalong

thestreamwisedirection.

Eq.16

ThetwophasefrictionfactorisafunctionofthetwophaseReynoldsnumber: Eq.17Thereareseveraltwophaseviscositymodels,McAdams[13]proposedthefollowing:

1

1

Eq.18

2.1.1.3.1.2 SeparatedFlowModel(SFM)In separated flowmodelsgasand liquidareconsidered to flowapart fromeachother. In

these typesofmodelsaccelerationaland frictional termsare treated separately.The first

oneisexpressedintermsofpipeinletandoutlet:

, 1 ,1 , 1 ,

1 Eq.19WherethevoidfractionisdeterminedfromZivis[14]relation:

1 1 Eq.20OneofthemostpopularfrictionalpressuredropcorrelationistheoneproposedbyFriedel

[15]:

2

,

Eq.21


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2.2 FinitedifferencemethodsInthischapterdifferentbasicfinitedifferencemethodsareexplained.

Theprincipleofafinitedifferencemethodisthatthederivativesinthepartialdifferential

equationare

approximated

to

linear

combinations

of

function

values

at

the

grid

points.

So,

havingadomainandagridlikethefollowing:

0 , 0 , 1 , , , , 0, 1 , ,

Thefunctionvaluesforeachgridpointwillbe:

Firstorderderivativecanbedefinedwithanyofthesethreeequalities:

l im l i m

lim 2

2 Eqs.36,37,38

Resultingfromthatdefinition,threefirstderivativeapproximationscanbedone:

Theforwardfinitedifference 2 Eq.39

Therefore

the

absolute

error

of

the

forward

finite

difference

is

proportional

to

andtheapproximationisreferredasafirstorderapproximation.

Thebackwardfinitedifference 2 Eq.40

Whichisagainafirstorderapproximation.

Thecentralfinitedifference


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12

2 2 6 Eq.41Theerror

isproportionalto

sotheapproximationisreferredasasecondorder

approximation.Note

that:

12 Eq.42

Figure4 Geometricinterpretationofthedifferentapproximations

Approximationtohighordersderivativescanbeobtainedfromtheformulasforlowerorder

derivatives.Thecentralfinitedifferenceschemeforthesecondorderderivativeis: 2 12 Eq.43

Soitisasecondorderapproximation.

Theonedimensionheatequationwithasourceisanequationofinterestinthisthesis.Itisa

parabolicpartialdifferentialequation:

, Eq.44If isaconstantvalueand thepartialderivativesarepresented in theirshorterway, theequationcanbewrittenlikethis:

, Eq.45TheforwardEulermethod(forwardintime,centralinspace)is:

2

Eq.46


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13

0 , 1 , , 1 , 2 , , 1 ,wheretheboundaryconditions, and, aregiven,andwhere

isanapproximationto

,

and

,

.

Themaincharacteristicofthisschemeisthatthereisonlyoneunknownvaluewhichisthe

function value in the next time step for this grid point . Thatmeans that each newfunctionvalueineachgridpointcanbecalculatedapartfromtheothers.

However, to guarantee stability in a forward scheme the should satisfy the time steprestriction:

0 2

This time step restrictioncan implyabigdisadvantagewhen talkingaboutcomputational

timeconsuming.

The truncation error of this method is first order in time and second order in space.On theotherhand,thesameequationwritten inbackwarddifferencescheme (alsocalled

implicitscheme)is:

2 Eq.47Inan implicitschemetheonlyknownvalue is soa linearequationsystemneedstobesolvedforeverynewtimestep.

The truncation error is also . The good point of this backward Euler methodschemeisunconditionallystable,meaningthatcouldtakeanyvalue.Comparisonbetweenthesemethods

Theexplicit

or

forward

finite

difference

scheme

it

is

usually

easier

to

implement

because

it

goesoneequationbyoneequationandthatmakesitsimplytodevelop.However,thetime

step restriction isaharddisadvantagebecause itcancausehighcomputational resources

consumption.

Theimplicitorbackwardfinitedifferenceschemehasthereallyattractivepropertyofbeing

always stable, itdoesnotmatterwhich is thevalueof timestep.Thatpropertygives the

model a really robust behaviour which is a valuable characteristic when programming.

However, this scheme is more difficult to implement because of the matrix system that

needsto

be

created.


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14

3 Calculations3.1 CaseofstudyThisthesisfocusesinashellandtubemultistreamLNGheatexchanger.Specifically,onlyits

basicelement

is

going

to

be

studied.

A

multipass

shell

and

tube

MSHE

is

composed

of

hundredsofshelltubeswhileeachofthemhasastubesinsideasstreamsareleft.

Figure5 Shellandtwotubesmultistreamheatexchangerscheme

Since the dynamics of a multistream shell and tubes heat exchanger is a complex

phenomenon to model, a simpler case will be studied in this thesis. The dynamics of a

counterflowshellandtubeheatexchangerhavebeenmodelled.Inthetubesidenaturalgas

isliquefiedwhileintheshellsidemixedrefrigerantvaporizes.Althoughitisasimplercase,it

hasprovedtobeaconsiderablechallenge.


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15

Figure6 Shellandtubeheatexchangerscheme

Sinceflowmaldistributionispartoftheheatexchangersproblematicwherebetterdesigning

toolsareneeded,massflowinthismodelisnotafixedvaluebutdependsonthedifference

betweentheinletandoutletpressures.

Anotherimportantissuethathastobetakenintoaccounttoachievebetterheatexchanger

designs isfluidpropertiesdependenceonstate. Inthemodel,fluidpropertiesaregivenby

TP_library,which isathermodynamicpackagecreatedbySINTEFRefrigerationEngineering

whichreturnsitsuserseveralfluidpropertiesforacertainthermodynamicstate.

3.1.1 Counterflowheatexchanger:NaturalgasliquefactionInthiscasethenaturalgasphasechangefromliquidtogasismodelled.

3.1.1.1 GeneraldataThecompositionofthenaturalgasusedinthemodelisthefollowing:

Table1Naturalgasandmixedrefrigerantscomposition

Naturalgas Mixedrefrigerant

MolefractionCH4 0.94480 0.29130

MolefractionC2H6 0.00580 0.38870

MolefractionC3H8 0.01890 0

MolefractionnC4H10 0.00110 0.22710

MolefractionN2 0.02950 0.09290

Criticalpressure[bar] 56 44.16


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16

SincethroughTP_librarycriticalpressuresarenotdirectlyobtainable,naturalgasandmixed

refrigerant critical pressures have been set at these approximate values. When accurate

resultsaredesired,thesevaluesshouldbecheckedandreplacedifnecessary.

Thegeometricaldataoftheheatexchangeris:

Table2Tubeandshellgeometricaldata

Tube Shell

Innerdiameter[mm] 4 10

Roughness[m] 1 1

Length[m] 2

Thetubematerialisaluminium.Itspropertiesareconsideredtobeconstantandare:

Table3 Aluminiumproperties

Density[kg/m3] 2700

Intensiveheatcapacitance[J/kgK] 930.77

Aluminiumheatconductivityisnotdisplayedbecauseitisassumedtobeinfinite.

3.1.1.2 StartingconditionsWhenthesimulationbegins,thenaturalgasinthetubesideisconsideredtobeinauniform

superheatedvapour

state

at

298K

and

49.2

bar.

Tube

wall

temperature

at

the

0 is297K through all its length. Finally,mixed refrigerant starting conditions are superheatedvapourat296Kand4.7barthroughalltheheatexchangerlength.

3.1.1.3 Boundaryconditions3.1.1.3.1 Tube sideNaturalgas inletenthalpy is theonecorresponding to superheatedvapourat298Kand it

keepsconstantduringallthesimulationtime.Outletpressure iskeptconstantat49.2bar

whileinlet

pressure

raises

from

50

at

0 till55barat 3 0 .


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17

Figure7 Naturalgasinputpressuredependenceontime3.1.1.3.2 Shell sideMixedrefrigerant inletenthalpyat 0 istheonecorrespondingtosuperheatedvapourat296Kbutitdecreasestillbeingsubcooledliquidat110Kat 6 0 .Once steady state conditions have been reached, mixed refrigerant inlet temperature is

raised again to observe the thermal dynamic behaviour of the heat exchanger. Inlet

temperature raises from 110K at

5 0 0 to 250K at

5 2 0 where ismaintained till

5 4 0 anddecreasedagainto110Kat

5 6 0 .

Figure8 Mixedrefrigerantinlettemperaturedependenceontime


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18

Inletandoutletpressuresfollowananalogouslawtonaturalgasonebutwithlowervalues.

Figure9 Mixedrefrigerantinletpressuredependenceontime

3.1.1.4 HeattransfercoefficientandpressuredropgradientdefinitionThecorrelationsselectedtobeusedinthemodelweredescribedinchapter6butthefinal

definitionthroughthatmakesthemcontinuousthroughallfluidstateshasyettobedone.

Single phase and two phase correlations need to be linked by some smooth functions,

hence,there

have

to

be

defined

some

vapour

fractions

limits

that

these

functions

will

link.

Here, the lower limit isvapour fractionequal to0.3and theupper limit isvapour fraction

equalto0.7.

Equationsthatdefinetheheattransfercoefficientarethefollowing:

Analogously,equationsthatdefinepressuredropgradientare:


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19

3.1.1.5 SimulationaccuracyThegridconsistsof40cellsandtheconvergencecriterionisequalto10

3.


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20

3.2 Themodel3.2.1 ProcedureDuringtheperiodthisthesishasbeendeveloped,thesameprocedurehasbeenfollowedto

createand

evaluate

different

models.

When

developing

amodel,

it

has

been

applied

the

fromsimpletocomplexrule,thatmeansthatfirstthewholestructureofthemodelwas

builtwithout taking intoaccountcharacteristics thatcould introducemore instabilities.So

firstapipewith fixedanddefinedwall temperaturewasmodelledwhere fluidproperties,

heattransfercoefficientandpressuregradientwereconsideredtobeconstant.Andstepby

step,morecomplexitywasintroduced.

Once the pipe model proved to work correctly with constant fluid properties, real

thermodynamicswere includedby theuseof TP_library.So, first themodelwas checked

withpure

water,

because

of

the

better

knowledge

of

the

thermodynamical

behaviour

of

this

fluid.Thenthemodelwascheckedwithpuremethaneastheworkingfluidwhichhasmuch

more similaritieswithnaturalgasbut it is still apure component.Finally, themodelwas

testedwithnaturalgaswhichasamulticomponent fluidwith theeffectson itsproperties

thatcarriesrepresentedahigherchallenge.

When real fluid properties hadbeen tested satisfactorily in themodel,heat transfer and

pressure drop correlations were included. First, heat transfer coefficient correlation was

includedwhile pressure dropwas set constant or following a linear rulewithmass flow.

Thentheoppositesituationwasfollowed,pressuredropcorrelationwasincludedwhileheat

transfercoefficientjustfollowedasimplerule.Finally,bothcorrelationswereincluded.

At theend,once thepipemodelhadproved togive satisfactory results, thecounter flow

heatexchangerwasmodelledfollowingthesamecalculationstructureasinthepipemodel.


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21

3.2.2 StructureandoperationThemodeldeveloped in this thesisallowssimulating thedynamicsofacounter flowheat

exchangerwhere the two fluids suffer a phase change. In thismodel themass flowof a

streamisnotdefinedbutdependsonthedifferenceoftheinletandoutletpressures.

Figure10 Discretizedschemeoftheheatexchangermodelled

TheprogramhasbeenwritteninMATLAB.

Inthismodelthestateofthefluid issetthroughitsenthalpyandpressure.Othervariables

liketemperatureandvapourfractionbecomeaconsequenceofthem.Thestateofthewall

isdefinedbyitstemperature.

Someassumptions

are

taken

in

this

model:

Stableflow. Homogenousflowregime. Thermodynamicequilibrium. Nonaxialheatconduction. Infinitewallthermalconductivity(walltemperatureisconsideredtobethesameon

bothsidesofonewallcell).

Therearesomeimportantcharacteristicsofthefinalandsatisfactorymodeldevelopedthat

shouldbementioned.Firstoneisthatenthalpiesandwalltemperaturescalculationsfollow


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22

an implicit scheme. The second one is that it presents a varying time step (between

established limits) depending on time derivatives. Also, a function that governs the

weighting coefficient value in solution relaxation has been implemented. Further

explanationofalltheseisgiveninthischapter.

Theoperationofthemodelfollowstheschemebelow:

1. Massflowandpressurescalculationofbothshellandtubestreams.2. Enthalpiescalculationofbothshellandtubestreams.3. Convergencecheckingonthestreamsobtainedsolutions.4. Relaxingstreamssolutions.5. Other streams variables calculation (temperatures, vapour fractions, heat flows,

etc.).A flashcallgives thenew temperaturesandnewvapour fractions, then,new

heattransfercoefficientsandnewheatflowscanbefound.

6. Newwalltemperaturescalculation.7. Convergencecheckingonthewallobtainedsolution.8. Relaxingwallsolution.9. Ifboth streams andwall convergenceswere achieved, resultsare storedandnew

time step calculationswill start. If convergencewasnot achieved, anew iteration

startsagainwiththerecentcalculatedvalues.


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23

Figure11 Modeloperationscheme


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24

It is asked the reader to note that it is an iterating scheme and that,during calculations

between time steps and , as many iterations as necessary will be done tillconvergenceisreachedforallcells.

3.2.2.1 PressuresandmassflowcalculationThetotalstreampressuredrop istheresultofthe integrationofeachdifferentialpressure

dropalongthefluiddirection:

Eq.50Whendiscretized,thetotalpressuredropbecomesasumofeachcellpressuredrop.Taking

intoaccountthattheothervariablesarekeptconstantduringthiscalculation,pressuredrop

dependsonlyonthemassflow.

Eq.51A combination of different numerical methods is used to find the mass flow rate which

correspondstotheboundarypressuredrop.Thesecantmethod,thefalsepositionmethod

andthehalving intervalsmethodareusedhere.First,acoupleof iterationsaredoneusing

the secantmethod to find the upper and the lower limits for the false positionmethod.

Then,acertainnumberofiterationsaredonewiththefalsepositionmethodusingitsrobust

butalso

fast

characteristics.

If

convergence

is

not

achieved

after

this

certain

number

of

iterations, the secant method is used to find the mass flow rate. The halving intervals

methodisusedonlywhenconvergencehasnotbeenreachedbytheothertwomethods.

3.2.2.2 EnthalpiescalculationThefollowingbalanceequationneedstobesolvedinthemodelforeachstream:

Eq.52

Ascheme

of

the

energy

balance

for

each

cell

in

the

tube

is

the

following:


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25

Figure12Tubesidecellenergybalancescheme

Whendiscretized,theimplicitschemeformoftheEq.50is:

Eq.53Wheresuperscriptsdenotetimeandsubscriptsdenoteposition.

However,heatflowdependsonseveralvariables(temperature,vapourfraction,...)thatwill

befoundwiththeflashcall,whichdependagainontheenthalpy.So,heatflowstakevalues

from

last

iteration

because

is

not

possible

to

directly

isolate

the

enthalpy.

Asithasbeenexplainedinthetheorypart,animplicitschemerequiressolvingasystemof

equations.Thelinearsystemthathereneedstobesolvedis:

MATLABpresentsa reallypowerfulcharacteristicswhenoperatingwithmatrices,soapart

from the robust characteristics of the implicit scheme itself, the program gives an extra

advantageinthispartofthemodel.


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26

3.2.2.3 SolutionrelaxationOncepressuresandenthalpieshavebeencalculated,newstateshavebeenfound.However,

sometimestheseresultswould leadtonumerical instabilities.Sonewstatescannotalways

be the results obtained in the calculation. In some cases it is needed to do a weighting

relativebetween

new

new

state

results

and

the

old

new

state

results,

this

is

called

relaxing

thesolution.

1 0 1 Eq.55Heredenotesstateandisanenthalpyandpressurecouple,istheresultofaniteration.istheparameterusedtodotheweighting.Inthismodeltheparameterishasnotafixedvaluebutdependsonthecalculation.

When iterating, thereare two situations typesof situations thatcouldoccur.First type is

thateach

iteration

gives

anew

state

that

it

is

closer

to

the

real

solution

than

the

one

before,

then it is a stable situation. Second type is that each iteration gives a new state that is

further from real solution than the one before. Both types are shown in the following

figures:

Figure13 Numericallystablesituation

Figure14 Numericallyunstablesituation


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27

Theusualstrategywhenrelaxingthesolutionistofixthevalueoftheweightingcoefficient

for all the situations. If the coefficient is low enough and/or the convergence criterion is

wideenough, thatbrings toastablesituation.However, thisstrategy isnotoptimalwhen

thesituationisstablefromthebeginning.Inastablesituation,aweightingcoefficientvalue

lowerthan

one

will

lead

to

ahigher

number

of

the

necessary

iterations

to

achieve

convergencethanwhenthevalueofitisone.

Figure15 Fixedcoefficientrelaxingsolutionscheme

Inthemodeldeveloped inthisthesis thecoefficienthasnotaconstantvaluebutchanges

dependingon the resultsof iteration.Actually,everynew time step the coefficient starts

havingavalueequalto

.Thevalueof

issetbytheuserandcannotbehigherthan1.In

thesimulation

which

results

are

showed

in

this

thesis,

this

starting

value

is:

0.5Aftereach iteration, resultsarecheckedand thecoefficient isdecreasedorkeptconstant

dependingonthem.Thelogicalrulethatgovernsthecoefficientdecreasingisbasedonthe

followingtwocharacteristicsofanunstablesituation(seefigure8):

1. Absolutedifferencebetweenstatesraises:

| | | |

2. Differencebetweenstateschangesitsigneveryiteration: 0 0

Ifanunstablesituationisdetected,theweightingcoefficientisreduced:

0 . 9 5 Followingthatmethodologythesolution isrelaxedonlywhenneed,soafastestandstable

solutionisachieved:


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28

Figure16Behaviouroftherelaxingthesolutionfunctiondevelopedandusedinthemodel

Theweightingcoefficienthasaminimumvaluethatissetbytheuser.

Asit

has

been

explained,

fluids

states

are

set

by

apressure

enthalpy

couple,

and

there

are

twostreamsandseveralcellsforeachstream.Therelaxingcoefficientusedineachiteration

tocalculatenewstateswillhavethesamevalueforallcellsandinbothstreams,thismeans

that if one cell is found to be in an unstable situation the relaxing coefficient value will

decreaseforallofthem.

3.2.2.4 FlashcallOnce the solution relaxation ends, pressures and enthalpies values have been definitely

established,whichmeansthatnewfluidstateshavebeenfound.Laststepistofindtherest

ofthe

variables

that

are

basic

for

the

calculus

or

may

be

interesting

for

the

analysis.

These

variables are: temperature, vapour fraction, heat flow, heat transfer coefficient, etc.

Temperature,vapourfractionandliquidandvapourmolarcompositionsarefoundthrough

theTP_librarypressureenthalpyflashcall.

Itcannumericallyhappenthatenthalpyvalueislowenoughthatitsrelatedtemperatureis

0K, if thathappensall the fluidpropertiesgive falsevaluesand thecalculusof themodel

crash.Therefore,aminimumoutputtemperaturewasset,iftheflashoutputtemperatureis

lowerthan60Kthecellisconsideredtobeinliquidstateat60K.

Similarsituationhappenswhentemperaturerisesabove1000K,sowhenthathappensthe

cellisconsideredtobeinvapourstateat1000K.

Apartfromflashoutputvariables,heatflowneedstobecalculated.Thedifferentialformof

theheatflowis:

, Eq.56Here

, , is the convection coefficient,

is thedifferential fluidwall contact

surfaceand

and

arethewallandthefluidtemperature.


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29

Whendiscretized:

Eq.57The convection coefficient is calculated by an internal function which includes all the

necessarycorrelations

which

are

linked

by

the

pertinent

linking

functions.

An

important

characteristic that has to be mentioned is that whether a condensing or an evaporating

correlation isused inthe twophaseareadependsonlyon the inletconditions. If the fluid

enterstheheatexchangerasliquid,thenanevaporatingcorrelationwillbeused.Ifthefluid

enters theheatexchanger as vapour, then a condensing correlationwill beused. That is

mentionedbecause, inacrossflowheatexchanger,heatflowdirectioncouldswap locally,

whichcouldmeanavapourfractionincreasingofatwophasemixturethatwasexpectedto

condense or, a vapour fraction decreasing of a two phasemixture thatwas expected to

evaporate.

3.2.2.5 TimestepfunctionTimestepinthemodelisnotconstant,itischangeddependingontimederivativesofsome

of thevaluesof thevariables in it.These variablesarepressure,enthalpy,mass flowand

temperatures. Although the first two are enough to define a state, the other ones have

proved to give an accuracy increasing when taking into account. A new time step is set

considering thatavariablecannotchange itsvaluemore thana fraction that issetby the

user. Then, considering this fraction and the variable timederivative, themaximum time

stepforeachvariableisfoundandtheminimumofallischosen:

, , , , ,

, , min,, ,, ,, ,

Minimum and maximum time step values are set by the user in order to prevent

unreasonablyshortorlongtimesteps.

Since pressure, enthalpy and temperature values depend on the cell, a previous cell

maximumtimederivativeselectionhastobedone.

Timestepiscalculatedwhennewtimederivativesarenotstillknown,solastvaluesareused

to calculate them. However, real new time derivatives may be different than the ones

before,sovariablechangeswillbedifferentthanpredicted.


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30

Figure17 Modeltimestepfunctionscheme

Although timederivativescanbedifferent thanpredictedno instabilitieswillbeprovoked

becauseofthat.Infact,theobjectiveofthisfunctiontodetailedresultsduringdynamicsbut

allowingfastresultswhensteadystateisreached.

3.2.2.6 LinkingfunctionsWhen heat transfer coefficient and pressure drop correlations are used, a link has to be

createdbetweenthesinglephaseandthe twophaseareas inordertoguarantee function

continuityand, then,numerical stability. Itwasalsoobserved thatjustcontinuitywasnot

alwaysenoughtoguaranteestabilityandthatitsderivativeneededtobealsocontinuous.In

order to achieve that objective, linking polynomial functions are used in that case. The

degreeof thesepolynomials is threeand theyachieve thecontinuityofboth the function

anditsderivative.

Thefirstofthesetwosmoothfunctionslinktheliquidsinglephaseareawiththe0.3vapour

qualitytwophasearea.Thesecondsmoothfunctionlinksthe0.7vapourqualitytwophase

areawithallvapoursinglephasearea.Theserelativelywiderangesofvapourqualitiesare

neededto

achieve

the

desired

characteristics

of

the

functions.

3.2.2.7 WallenergybalanceConsideringthatitisassumedthatwalltemperatureisequalonbothfacesofthepipeinthe

samecell,theenergybalanceequationthatgovernsthewalltemperatureisthefollowing:

, , , Eq.58Where

denoteswalland

1and

2denoteshellandtubefluidsrespectively.

WhenEq.(50)isdiscretizedfollowingtheimplicitscheme:


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31

,, , , , , , , Eq.57WalltemperatureoftimestepcanbeisolatedfromEq.(51):

, , , ,, ,, , ,, Eq.59


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32

4 Resultsanddiscussion4.1 EvaluationofthemodelrobustnessandtimeconsumptionThemainobjectiveofthisthesiswastocreateanumericallystablemodel.Therefore,allthe

functionsused

in

it

have

been

analysed.

Linking

functions

have

been

created

when

it

has

beennecessarytoavoiddiscontinuitiesthatcouldcarrynumericalinstabilities.Thisstudyof

thenumericalbehaviourofthemodelhasrevealedsomefactsthataredescribedbelow.

4.1.1 PhasechangeathighpressureSinceLNGisproducedathighpressuresjustbelowitscriticalpoint,whichinthisthesishas

beensetat56bar,itwasfirstdecidedthatfinalinletandoutletpressuresduringsimulations

wouldbe55and54.2barrespectively.However,thesesimulationsbroughtalwaysa

numericalinstabilitywhoseoriginwasfinallyfoundandisexplainedhere.

Ithasbeenobservedthatthenaturalgasphasechangeathighpressure isaphenomenon

where vapour fraction varies abruptlywith enthalpy. Besides, it has been noticed that it

could be considered a function discontinuity when pressure is high enough. That

discontinuityhas severeconsequencesonother functionsof themodel, likeheat transfer

coefficientone,andthatprovokesanirresolvablenumericalinstability.

Thefollowingfiguresshowthevapourfractionandheattransfercoefficientdependenceon

enthalpy at different pressures.Distance between dots in the phase change is 100 kJ/kg

whichisashortintervalconsideringtheenthalpymagnitude.

Resultswhenpressureisequalto50bar:

Figure18 Vapourfractionvs.Enthalpyat50bar


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33

Figure19 Heattransfercoefficientdependencevs.Enthalpyat50bar

Resultswhenpressure isequal to54bar (here,dotsenthalpy intervalwhen 0 0.1wassetequalto1kJ/kgtoassurethattherewasadiscontinuity):



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34

Figure21 Heattransfercoefficientvs.Enthalpyat54bar

Resultswhenpressure isequal to55bar (here,dotsenthalpy intervalwhen 0 0.6wassetequalto1kJ/kgtoassurethattherewasadiscontinuity):



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35

Figure23 Heattransfercoefficientvs.Enthalpyat55bar

Noticethatat54bar italreadyappearsadiscontinuityat lowvapourfractions.However,at

55barthisdiscontinuitygoesfrom 0to 0.58,andthatprovokesthatheattransfersuffersalsoadiscontinuity thatgoesalmost from itsminimum value to itshighest value,

whichwouldbeafocusofenormousnumericalinstabilities.

Sincevapour

fraction

correspondence

with

enthalpy

depends

on

the

flash

call,

and

this

one

depends itself on the TP_library, there is no possibility of avoiding this vapour fraction

discontinuity without losing all the advantages and flexibility that TP_library allows.

Therefore, it is decided to reduce the final inlet pressure to 50 bar,which is also a high

pressurewhereinphasechangevariablespresentabruptlyvariationsbutiscontinuous.

4.1.2 GnielinskiandDittusandBoelterheattransfercorrelationsnumericaleffects

Another relevantobservation thathasbeennoticedduring simulations is that correlation

electioncan

have

an

important

impact

on

the

numerical

oscillations

created

during

calculations,hence,onCPUtimeconsumption.

Two simulations were run to compare Gnielinski and Dittus and Boelter heat transfer

correlationsnumericaleffects.All theparametersare the sameasdescribed in chapter8

except from some accuracy parameters that have reduced to increase simulation speed.

Thesechangesare:

Grid:Thecellsnumberhasbeenreducedto20. Convergencecriterionissetat103. Timestepfunction: 1.


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36

So, once these modifications were done, both simulations were run until 4 0 . Thefollowingfiguresshowthenumberof iterationsnecessarytosolveeachtimestepforeach

simulation:

Figure24 NumberofiterationsnecessarytoachieveconvergencewhenGnielinskicorrelationisused

Figure25 NumberofiterationsnecessarytoachieveconvergencewhenDittusandBoeltercorrelationisused

ThemeannumberofiterationsusedinthesimulationwhereGnielinskicorrelationhasbeen

used is15,175 iterationspersimulatedsecond,while in theonewhereDittusandBoelter


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37

hasbeenusedis43,125.BothcorrelationswouldbevalidtouseinthemodelbutGnielinski

correlationprovedtogiveafastersolution,sothisisthecorrelationusedinthemodel.

4.1.3 AdvantagesofthevaryingrelaxingcoefficientSince

a

function

was

developed

just

to

control

the

relaxing

coefficient

value,

it

was

necessary

toproveitsusefulness.

Same simulation parameters as the ones used to compare correlations were used in a

simulationwithafixedrelaxingcoefficientvalue(andusingGnielinskicorrelation).

The relaxing coefficientwas fixedat0.2and thenumberof iterationsnecessary to reach

convergenceeachtimeisshowedinthefollowingplot:

Figure26 Numberofiterationsnecessarytoachieveconvergencewhenafixedrelaxingcoefficientvalueisused

The mean number of iterations when the coefficient is constant is 27,6 iterations per

simulated second, while when it varies is only 15,125 (as showed in figure 24). So, the

functiondeveloped

to

control

the

value

of

this

coefficient

proves

to

give

afaster

solution.

4.1.4 EffectoftimesteplengthandgridmeshingonthenumericalstabilityIn linearnumerical implicitschemewhereparameters in ithaveaconstantvalue likeones

explainedinchapter7,timesteplengthdoesnotjeopardizenumericalstability.Asexplained

in chapter 7, only the explicit scheme is under a time step length restriction to avoid

instabilities.

However,themodeldevelopedinthisthesisisafollowsanonlinearimplicitscheme.Lotsof

parameters

vary

with

the

state

(like

fluid

properties).

As

a

consequence,

it

has

been

observedthattimesteplengthdoescompromisethenumericalstabilityofthemodel.Asan


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38

example,samesimulationastheoneheresimulatedwithaminimumtimestepof5sinstead

of2sanditconductedtoanumericalinstabilitythatmeantthefailureofthesimulation.

Similarconsequencehasbeenobservedwhentalkingaboutgridmeshing.Samesimulation

runwith20cellsasgridnumber,insteadof30,producedaseverenumericalinstability.

Itisreasonabletothinkthatbigcellsunderbigchangeswillproducemoreoscillations,and

sometimesinstabilities,thansmallercellsundersmallerchanges.


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39

4.2 CaseresultsThischaptershowtheobtainedresultswhensimulatingthecasepresentedinchapter3.1.

Case of study boundary conditionswere set in order to study the behaviour of the heat

exchangerduring

start

up

conditions

and

during

an

abrupt

change

in

the

inlet

mixed

refrigeranttemperature.

Itisremindedthatduringthestartup,bothmixedrefrigerantandnaturalgasinletpressures

increase till 6 0 , and from then on are kept at a constant value. Natural gas inlettemperatureiskeptconstantat298Kduringallsimulationtime.However,mixedrefrigerant

inlettemperaturefollowslawdependingontimethatisshowedinthefigurebelow:

Figure27 Mixedrefrigerantinlettemperaturedependenceontime

Whentheresultsofastreamareshownforseveralcells,acodeofcoloursisusedtoknow

thepositionofthesecells.Thiscodeistheonerepresentedinthenextfigure:

Figure28 Codeofcoloursusedwhenplottingdifferentcellsofasamestream


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40

Figure29 Mixedrefrigeranttemperatureinseveralshellcells

Figureaboveshowsthetemperaturesduringthesimulatedtimeofseveralcellsintheshell

side.Someinterestingissuescanbeexplainedfromit.

First of all, it shows how the start up conditions last till after

1 0 0 although mixed

refrigerant inlet temperature was kept constant from 6 0 . So, the system needsapproximately40stoachievesteadystateconditions.Figure 29 also shows the effects of the abrupt change in the mixed refrigerant inlet

temperature(seefigure27).NotethatMR inlettemperature iskeptconstantduring20sat

220Kbuttheheatexchangerstillrespondstoitasifitwasapeakinput.

Focusingonthenumericbehaviouronthemodel,animportantcharacteristicthatfigure30

illustrates is the good response of the time step function. Note that time step length

decreaseswhen

the

system

is

working

in

transient

conditions

and

as

aresult

detailed

information is obtained from themodel. Besides, it can also be observed that time step

length increaseswhen steady stateconditionsare reachedand thatmeans lessCPU time

consumption.

Sincethesystemreachessteadystateconditionsat 1 0 0 , fromthenonthe timesteplength is 30s. That could have provoked the lost of much of the information about the

abruptchangeconsequencesthatoccursat 5 0 0 .Toavoidthis,themodelisforcedto,first, reach exactly

5 0 0 and then, in the next calculation use the time step with

minimum length.Thiswayofsolving thisproblematic iscalledeventhandlingandsome

programslikeordinaryequationssolversinMATLABusethesameprocedure.


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41

Naturalgasmassflowrateduringthesimulatedtimeisshowninthefollowingfigure:

Figure30 Naturalgasmassflowrateduringtime

In figure30 itcanbeobservedagainthedynamicbehaviourofthesystem,here,NGmass

flowrateduringthesimulatedtime. It is interestingtofocus inthedecreasethatthemass

flowratesufferswhentheMR inlettemperaturechanges increasesabruptly.Thecauseof

thatphenomenon

is

that

natural

gas

outlet

phase

changes

from

liquid

to

vapour.

When

the

coldstream increases itstemperaturemoreandmorenaturalgas invapourstateemerges

fromthetubeoutlet,consideringthatpressuredropisconstantfrom 6 0 ,thisincreaseinthevapourfractionprovokesareductioninthemassflowrate.

MRmassflowratedecreasesasaconsequenceofthesamephenomenon.MRinletvapour

fractionincreaseswiththeinlettemperature.Consequently,massflowrateisreducedwith

inlettemperatureincreasing.

4.2.1 ThermaldynamicsduringthestartupOneofthemostappreciatedqualitiesofthismodelwillbeitscapabilityofsimulatingstarts

upandshutdowns.Therefore,detailedthermalresultsofthestartupinthissimulationare

givenbelow.


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42

Figure31 Naturalgascellstemperaturesduringt


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43

Figure33 Temperaturesandvapoursfractionsatt=20sandatt=40s

Figure33showsNG,wallandMR temperaturesalongthe lengthoftheheatexchanger in

twodifferenttimesteps,at

2 0 andat

4 0 . Italsoshowsvapour fractionofeach

streamat

the

specified

time

steps.

Figure33clearly illustratesthe influenceofthephasechange inthewalltemperature. It is

reminded that NG, wall and MR starting temperature were 298K, 297K and 296K,

respectively,so,bothstreamsstartingconditionsweresuperheatedvapour.However,both

frames show that wall temperature reduces considerably its difference with MR

temperature when two phaseMR flows through the shell side. The cause of that is the

increasingoftheheattransfercoefficientvalue;atwoheattransfercoefficientvalueismuch

higherthanthesinglephaseone.


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44

Figure34 Temperaturesandvapoursfractionsatt=44sandatt=100s

Figure34showsagaintwoframesofthestartup.Theseonesbelongtothesystemstateat 4 4 andat 1 0 0 (steadystate).First feature that has to be highlighted is the rapid NG phase change. As it has been

explainedinchapter7.1,NGphasechangeathighpressuresisaabruptphenomenon.Here,

itcan

be

noticed

how

it

became

all

liquid

in

just

4s

(comparing

with

figure

33

at

4 0 ).Second feature that should be commented about the first frame is that this liquid NGachieveshigherheattransfervaluesandthattightsthewalltemperaturetohighervalues.

Thiseffectisillustratedintherisingofthelastcellstemperature.

Third and last important displayed characteristic that should be discussed refers to the

steadystateconditionsat 1 0 0 .ItcanbeobservedthatwalltemperatureisonlyclosertoNGtemperaturewherethephasechangeoccurs.Inthepreviousmeterswalltemperature

is almost the same asMR temperature.After the phase change,wall temperature drops

againto

amiddle

value

within

both

streams

temperatures.

This

feature

is,

again,

consequenceoftheheattransfercoefficientvaluedependingphasearea.

Moredetailedframesaboutthestartuparedisplayedintheappendix.

4.2.2 ThermaldynamicsduringanabruptchangeintheshellinlettemperatureThesesecondresultsshowthethermaldynamicbehaviouroftheheatexchangermodel

whenjustoneinletparameterismodified.


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45

Asexplainedinchapter6.1anddisplayedinfigure27,shellinlettemperaturesuffersarapid

change between 500s and 560s. This change in the inlet conditions has an effect in the

wholeheatexchangerthatisinterestingtostudy.

Figure35 Naturalgascellstemperaturesduring 500s


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achievesastablestateduringthistimewhiletherestofthesystemisinconstantlychange

duringit.

Figure35showshowtheeffectsofthisabruptlychangearerapidlydampedinthetubeside.

Noticethat,

since

the

model

takes

into

account

the

fluid

movement,

there

is

adelay

betweentheincreasingoftheMRinletandoutlettemperatures.

Aninterestingfigurethatshowstheconsequencesoftakingintoaccountthisfluid

movementisdisplayedbelow:

Figure37 NG,wallandMRtemperaturesandNGandMRvapourfractionsatt=516sandatt=520s.Crosstemperatures

phenomenoncanbeobservedatt=520sframe

Figure37showsstreamandwallstatesat 5 1 6 andat 5 2 0 .Thealreadyexplainedeffectsofthephasechangeonthetemperaturescanbeobservedagain.However,sincethe

model takes into account fluidmovement, some interesting features canbeextractedof

them.First,itcanbeseeninbothframesthatthereisaminimumat

9 inthevapour

fractionthatcanonlybenoticedbecausefluidmovementeffectisnotneglected.

Notice that at 5 2 0 MR temperature at shell inlet rises above NG and walltemperatures. That means that there cold and warm streams swapped locally. That is a

consequence of taking into account the fluid movement. Since it is a counter flow heat

exchanger,NGisfacingMRoutlettemperatureatitsinletandthatprovokesthatNGoutlet

temperaturecanbe lower thanMR temperaturewhenMR inlet temperature is increased

rapidlyenough.

However,the

model

does

not

take

into

account

this

cold

and

warm

streams

swapping

when

calculatingheattransfercoefficientvalue.ThismeansthatMRtwophaseheattransfervalue


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when itislocallywarmerisstillcalculatedthroughanevaporationcorrelation,whichisnot

correct.However,ithastobeconsideredthatchangingheattransfercoefficientcorrelation

followingheatflowdirectionwouldprovokeadiscontinuityintheheattransferfunctionthat

wouldtriggerasevereinstability.

More frames about walls and streams states during this period are displayed in the

appendix.


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5 Conclusionsandrecommendationsforfurtherwork5.1 ConclusionsDuringthedevelopmentofthisthesissomeinterestingnumericalbehavioursrelatedtoheat

exchangerwhere

phase

change

occurs

have

been

observed.

It

has

been

proved

that

if

the

pressureatwhichthephasechangetakesplaceishighenough,itcancreateadiscontinuity

in the vapour fraction dependence on enthalpy. This discontinuity affects severely the

numeric behaviour of themodel, it provokes discontinuities on other functions like heat

transfercoefficientone,andthatobviouslyleadstoanumericalinstability.

A second observation noticed when simulating in this thesis is the importance of the

correlations selection.Whetheroneorother correlation is chosenhas an impactonCPU

timeconsumption.Forthesameinputsimulationparameters,ithasbeenprovedthatDittus

andBoelter

single

phase

heat

transfer

correlation

leads

to

ahigher

number

of

necessary

iterationstoachieveconvergencethanGnielinskis.

Anotherrelevantoutputofthisthesisisthedevelopmentofasimplefunctionthatvariesthe

weightingcoefficientvaluewhenrelaxingthesolution.Thisfunctionhasbeentestedagainst

acoefficientwithafixedvalueandithasprovedtoproduceafastersolution.

Ithasalsobeennoticedthattimesteplengthandcellsizehaveanimpactonthenumerical

instabilityofthemodel.

Ithas

been

shown

that

the

model

results

provide

detailed

information

of

the

heat

exchanger

dynamics.Timestepcontrolcanbeanappreciated featureof themodel.Also,accounting

fluidmovementhasshownsome interestingresultsthatwouldhavebeenskipped if ithad

beenneglected.

Theobjectiveof this thesiswas to findout if it ispossible todevelopanumericallystable

multistreamheatexchangermodelwherethermaldynamicscouldbesimulatedandstudied.

Pursuingananswer to thisobjectiveastablecounterflowheatexchangermodelhasbeen

satisfactorilydeveloped.Thismodelaimstobeasuitablebasetocontinue inthedesignof

the

MSHE

model.


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5.2 RecommendationsforfurtherworkRecommendationsforfurtherworkcanbeclassifiedintothethreefollowinggoals:

1. Optimisationofthenumericalpartofthemodel.2. Researchofthesuitableheattransferandpressurecorrelations.3. Introductionofthedevelopedtoolintopracticaluse.

The model developed in this thesis has still to overcome some numerical challenges till

presentingthesuitablecharacteristicstobeusedasaheatexchangerdesigntool.Stabilityof

thesimulationsmustbeguaranteed,hence,furtherstudyonthetimestep lengthandgrid

meshingshouldbeconsidered. It isthewriteropinionthat improvedrelaxingsolutionand

time step functions could give much more robustness to the model. Besides, the

improvementofthesefunctionscouldalsomeanareductionintheCPUtimeconsumption,

whichisthenumericalchallengethatshouldbefacedoncestabilityisguaranteed.

Furtherresearchshouldbedoneinthecorrelationsfield.Findingthecorrelationsthatbetter

correspond to theexperimentalresultshasnotbeen thegoalof this thesis.Therefore,an

evaluation of the different available correlations and its accuracy should be carried out.

However,italsohastobetakenintoaccountthenumericconsequencesthatthiscorrelation

maybring.Itisremindedthatithasbeenshowninthisthesisthatcorrelationselectionhas

aneffectonthemodelnumericalresponse.

Finally,oncetheseand futurechallengeshavebeenconquered,themodelshouldbeused

forthe

real

design

of

aheat

exchanger.

Model

flexibility

during

the

design

process

and

the

accuracyoftheresultsmaybethemainappreciatedfeatures.


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6 Bibliography1. Adrian J. Finn, G.L.J., Terry R. Tomlinson, LNGtechnologyforOffshoreandmid-scaleplants.

2000.2. Julio Cesar Paco, C.A.D.,Areviewonheatexchangerthermalhydraulic

modelsfor

cryogenic

applications. 2011.3. Fredheim AO, H.R., Possibilitiesofcostreductionsinbase-load.EUROGAS96. Proceedings from the European applied researchconference on natural gas, 1996: p. 10114.4. Kanoglu M, D.I., Rosen MA, Performanceanalysisofgasliquefaction

cycles. Int. J. Energy Res, 2008.5. RF, B., Cryogenictechnology. Ullman's encyclopedia of industrialchemistry. WileyVCH Verlag GmbH and Co, KGaA, 2000.6. Warren M. Rohsenow, J.P.H., Young I. Cho, ed.

Handbookofheat

transfer. 1998.7. F. W. Dittus, B., Heattransferinturbulentpipeandchannelflow. 1930.8. Gnielinski, V., Newequationsforheattransferinturbulentpipeandchannelflow. Int. Chem. Eng, 1976: p. 359368.9. Stefan S. Bertsch, E.A.G., Suresh V. Garimella,Acompositeheattransfercorrelationforsaturatedflowboilinginsmallchannels. Int. J. HeatMass Transfer, 2009. 52: p. 21102118.

10. Z. Liu, W.,Ageneralcorrelationforsaturatedandsubcooledflow

boilingintubesandannuli,basedonanucleatepoolboilingequation.

1991.11. Shah, M.M.,Ageneralcorrelationforheattransferduringfilmcondensationinsidepipes. 1978.12. Issam Mudawar, J.L., Two-phaseflowinhigh-heat-fluxmicro-channelheatsinkforrefrigerationcoolingapplications:PartIpressuredrop

characteristics. 2004.13. W. Qu, I.M., Measurementandpredictionofpressuredropintwophasemicro-channelsheatsinks.

Int. J. Heat Mass Transfer, 2003.46

: p.27372753.14. Zivi, S.M., Estimationofsteady-statevoid-fractionbymeansoftheprincipleofminimumentropyproduction. ASME J. Heat Transfer 1964.86: p. 247252.15. J.G. Collier, J.R.T., Convectiveboilingandcondensation. third ed.,Oxford University Press, 1994.16. F. P. Incropera, D.P.D., FundamentalsofHeatandMassTransfer,fifthed. 2002.17. R. K. Shah, A.L.L., LaminarFlowForcedConvectioninDucts:ASourceBookofCompactHeatExchangerAnalyticalData.

1978.


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7 AppendixDetailedresultframesduringsimulationsstartupperiod:

Temperaturesandvapourfractionsatt=8sandatt=16s



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SomeframesduringtheabruptchangeoftheMRinlettemperatureare:




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Documents

Modelling of Multistream LNG Heat