超临界CO2热力性质快速计算方法

第第页超临界CO2热力性质快速计算方法FastCalculationMethodforSupercriticalCO2

ThermodynamicProperties

ZhaoDan，DingGuoliang，WuZhigang

InstituteofRefrigerationandCryogenics，ShanghaiJiaoTongUniversity，Shanghai（202340）E-mail：danzhao@，glding@，wzg@

Abstract

Calculationsofrefrigerantthermalpropertiesaredesiredtobeveryfastandstableincasofsimulationofrefrigerationsystem,etc.Theimplicitcurve-fittingmethodisaquitegoodmethodforthefastandstablecalculationofrefrigerantthermodynamicproperties,andithasbeenudforsaturated,two-pha,superheatedandsubcooledrefrigerants.Inthispaper,newfastcalculationmethodforsupercriticalCO2isprented.TheresultsshowsthatthecalculationspeedsofthefastcalculationformulaeforsupercriticalCO2aremorethan100timesfasterthanthoofREFPROP7.1whilethetotalmeanrelativedeviationofthefastcalculationformulaefromREFPROP7.1islessthan1.8%.

Keywords:CO2；Fastcalculation；Thermodynamicproperties；Supercritical

1.Introduction

Sincesimulationisappliedindesigningrefrigerationproductsmorewidelythaneverbefore,moreattentionsshouldbepaidtodevelopthemethodsforthecalculationofrefrigerantthermodynamicpropertieswhichneedverystrictrequirementsonstabilityandspeed.[1]TheEOS(equationofstate)methodisusuallyudtopredictrefrigerantthermalpropertiesinawiderangewithhighprecision.[2-8]Butthecalculationspeedandstabilityarelimitedbyunavoidableiterationsincalculation.Theexplicitpolynomialregressionmethodisanothersimpleyetfastcalculationmethodforrefrigerantthermodynamicproperties,andthestabilityofthismethodisalsobetterthanEOSwhiletheaccuracyisstillsatisfied[9,10].Cleland[11,12]prentedamethodtospeedupthecalculationofrefrigerantthermodynamicpropertiesandgavethecorrelationsforR12,R22,R114,R502,R717(NH3)andR134aforthesaturationtemperatureof-60°C~60°C.Butthismethodcannotguaranteethecalculationreversibility,andsodivergencemighthappeninsimulationunlessextremelyhighregressingaccuracyisapplied.

Inordertoensurecalculationreversibilityofrefrigerantthermodynamicpropertiesbesidescalculationrapidnessandstability,Dingetal.[1,17]developedtheimplicitcurve-fittingmethod.Theimplicitcurve-fittingmethodissuitableforbothpurerefrigerantsandrefrigerantmixturesandithasb

eenudtopredictthethermodynamicpropertiesforsaturatedliquid,saturatedvapour,twopha,superheatedandsubcooledrefrigerants.Thismethodcanbedividedintotwosteps:(1)constructtheimplicitcurve-fittingmodelwhichdependsonrefrigerantpha;(2)getcorrelationsforcalculatingrefrigerantthermodynamicproperties,namely,explicitfastcalculationformulae,bysolvingthestandardcubicequationwhichistransformedfromtheimplicitcurve-fittingmodel.Twoanalyticalsolutionsfromthesameimplicitcurve-fittingmodelcanensurereversibility.However,theimplicitcurve-fittingmethodforthesupercriticalrefrigerantsisnotreported.Andallthepublishedfastcalculationmethodscannotbeudinthesupercriticalregion.

NaturalrefrigerantCO2isoftenudundertheworkingconditionthatthepressureishigherthanthecriticalpressure.Forexample,supercriticalCO2oftenworksintheheatpumpandautomobileairconditioning.Thereforeanewimplicitcurve-fittingmethodtopredictsupercriticalCO2thermodynamicpropertiesisnecessary.

Thepurpoofthispaperistoprentanimplicitcurve-fittingmethodtocalculatethethermodynamicpropertiesforsupercriticalCO2.ThecalculationresultsofthefastcalculationformulaeofCO2willbecomparedwiththedataproducedbyREFPROP7.1inordertoshowthecurve-fittingaccuracy,andthecalculationspeedofthefastcalculationformulaeofCO2willbecomp

aredwiththatofREFPROP7.1.

SpecializedRearchFundfortheDoctoralProgramofHigherEducation(GrantNo.20230248019)NationalNaturalScienceFoundationofChina(GrantNo.50576053)

2.Curve-fittingmodelforthefastcalculationofsupercriticalCO2

thermodynamicproperties

Fortheimplicitcurve-fittingmethod,theexplicitequationsshouldbetheanalyticalsolutionsofanimplicitpolynomialequation,namely,implicitcurve-fittingmodel.Soitisveryimportanttoconstructthecurve-fittingmodelforthefastcalculationofsupercriticalCO2thermodynamicproperties.Thecurve-fittingmodelshouldbebadonthecurvefigure,andFig.1showsthepressure-enthalpychart.

Fig1pressure-volumechart

Thecurvesinthechartaretheisothermallineswithuniformdistribution.Wecanobrvethatthedifferencebetweenenthalpyvaluesontheadjacentisothermallinesunderthesamepressureisalmostthesame.Sowecanget

001

k

AAkC≅(1)Wherekistheproportionalitycoefficient,Cisaconstant.Martin-Houequationcanbewrittenasfollowingform

23231234567823

()()aaTaTaTaaTaTaTRTtyrannosaur

pvbvbvb++++++−=+−−−(2)

FortheisothermallineswithtemperatureT0,equation(2)canbewrittenasfollowingform

bar是什么意思>四级多少分可以考口语

2323

01203040560708023000()()RTaaTaTaTaaTaTaTpvb

vbvb++++++−=+

−−−(3)Sinceconstantbisdifficulttoregressbytheleastsquaremethod,soweneglectit.Underthesame

福州日语培训pressure,equation(3)minusequation(2)is

23230000123422222222000002

3

2

3

000567822222222

000011

()()()()(11

(

)()()()0TTTTTTTTRaaaavvvvvvvvvvTTTTTTaaaavvvvvvvv−+−+−+−+−+−+−+−+−=(4)

Theequationjustreflectstherelationshipmentionedabove.SuppotemperatureT0isaconstantand

itsvalueishigherthancriticaltemperature.FortheisothermallineswithtemperatureT0,v0onlydependonpressureP,andv0isafunctionofpressureP,thatisv0=f(P).Thefunctioncanbeobtainedbyusingtheimplicitcurve-fittingmethodforsuturedCO2thermodynamicproperties[13].Sincespecific

P0

volumev0canbecalculatewhenpressurePisknown,theequation(4)istheimplicitcorrelationabouttemperatureTandspecificvolumev.

Weobrvethatnotonlyspecificvolumehavetherelationshipundersamepressure,otherthermalpropertiesalsodoes.Forexample,theenthalpyalsohastherelationshipasshowninfig2.Therefore,itisreasonablethattheequation(4)canalsobeudastheimplicitcorrelationabouttemperatureTandotherthermalpropertieswhenthespecificvolumevisreplacedbyotherthermalproperties.doss

simple是什么意思

Fig2pressure-enthalpychart

Inordertoimprovetheprecisionoftheequation,someloworderitemscanbeaddedtotheequation(4)andtheimplicitcurve-fittingmodelforthesupercriticalCO2canbeobtainedasfollows.

2323000123000023230004567222222220000232300089101133333333000011

()()()()11

()()()()

11

()()()(0

TTTTTTaaauuuuuuuuTTTTTTaaaauuuuuuuuTTTTTTaaaauuuuuuuu

−+−++−+−+−+−+−+−+−+−+−+−=(5)

where,a1,a2,…a11arethecoefficientstoberegresd;uisasimplefunctionofz.Forexample,u=

z/100,etc.Itcanbewrittenasu=f(z),andzisoneofsupercriticalthermalpropertiesexcepttemperatureTandpressureP,suchasenthalpy,entropy,density,etc.Afterthetransformofthethermalproperties,theprecisionofthemodelcanbeimproved.T0isaconstantanditsvalueishig

herthancriticaltemperature.u0isasimplefunctionofz0,thatisu0=f(z0).FortheisothermallineswithtemperatureT0,z0onlydependonpressureP,soz0isafunctionofpressureP,thatisz0=f(P).Thefunctioncanbeobtainedbyusingtheimplicitcurve-fittingmethodforsaturatedrefrigerantsthermodynamicproperties[13].

ConsideringtheusualapplicationconditionsofsupercriticalCO2,theexpectedapplicationrangeofthefastcalculationformulaeis:temperaturein304.15K～393.15K;pressurein7.3773MPa～12.0MPa(thecriticaltemperatureandcriticalpressureforCO2are304.15Kand7.3773MPa,respectively.)ThedatasourceforregressingcomesfromNISTREFPROP7.1[2].

Fortheimplicitcurve-fittingmodelequation(5),wetT0=318.15K.Thecoefficientsa1,a2,…a11canberegresdbyusingthedatasourcefromNISTREFPROP7.1[2].Table1liststhecoefficientsforimplicitcurve-fittingmodel.

P

Table1CoefficientsofimplicitequationsforthermodynamicpropertiesofsupercriticalCO2Implicitform

f(p,T,s)=0f(p,T,h)=0f(p,T,ρ)=0f(p,T,λ)=0f(p,T,µ)=0ush1/7

5-ρ/1000100-100λ(106µ)1/6a1

-3.8341×10-3-4.9998×10-4-8.7673×10-7-6.3423×10-65.2334×10-3a2

1.9232×10-51.9449×10-61.7061×10-71.5205×10-7-3.2281×10-5a3

-2.5091×10-8-6.3650×10-10-5.1530×10-10-3.7780×10-105.0088×10-8a4

-1.5015×100-2.3208×100-4.5585×100-9.5058×101-1.7590×100a5

1.2500×10-2

2.3283×10-37.0312×10-4-1.5100×10-3-1.6951×10-2a6

-6.7218×10-5-7.9691×10-6-4.8476×10-6-1.0032×10-51.0686×10-4a7

9.1176×10-85.5420×10-108.2396×10-93.8006×10-8-1.6892×10-7a8

7.5499×10-11.7956×1006.9332×1003.0122×1031.0341×100a9

-1.1733×10-2-2.4629×10-3-3.4146×10-31.8826×10-11.3926×10-2a10

6.4670×10-5

7.1942×10-61.9655×10-5-3.9069×10-4-

8.9374×10-5a11-8.8767×10-83.0685×10-9-2.7975×10-8-2.0877×10-71.4361×10-7

3.Getexplicitfastcalculationformulafromimplicitcurve-fittingmodelforsupercriticalCO2thermodynamicproperties

WhenTisknown,let

23200000123456222

00000003230000789101123333000002312323456723

89101111

(1

)

广西会计从业资格考试试题1.0TTTTTAaaaaaauuuuuuuTTTTaaaaauuuuuBaTaTaTCaaTaTaTDaaTaTaT=−+++++++++++=+++=+++=+++(6)

thenEq.(5)becomes

023=+++DCuBuAu(7)

Aisanexplicitfunctionofpbecauu0isexplicitfunctionsofp,andB,CandDareexplicitfunctions

ofT.ThesolutionsofEq.(7),providedaccordingtoAppendix,arethereforeexplicitfunctionsofpandT,whichcanbeexpresdasu=u(p,T).Beingzisasimplefunctionofu,theexplicitequationz=f(p,T)canbegottenfinally.Forexample,suppohisthevariableneedtobecalculated,andu=h/100.Aftergettingtheexplicitequationu=u(p,T)bysolvingEq.(7),h=100u(p,T),whichistheexplicitformofh=f(p,T),canbeobtainedfinally.Whenzisknown,.Let

21173210622

951232

0008412342

00000滑铁卢大学专业

2323300000056789101122233330000000

11

(1

Aaauau

Baauau

CaauauTTT

DaauuaaaauuuuuTTTTTTaaaaaaau

uuuuuuu=++=++=++=++−+++++++++++(8)

thenEq.(5)becomes

023=+++DCTBTAT(9)

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where,Disexplicitfunctionofpbecauu0areexplicitfunctionsofp.A,BandCareexplicitfunctionsofu.BysolvingEq.(9)withthemethoddescribedinAppendix,threeroots,whichareexplicitfunctionsofT,canbeobtained.Afterchoosingtherightoneofthem,theexplicitformulaT=f(p,z)canbegotten.

Withtheabovemethod,thefastexplicitcalculationformulaecanbeobtainedandtable2liststheexplicitcalculationformulaeforthermodynamicpropertiesofsupercriticalCO2

Table2ExplicitcalculationformulaeforthermodynamicpropertiesofsupercriticalCO2formformA,B,CandDofcubicequation(4)（yistherootof

equation5）s=f(p,T)(6)(A5)

s=yT=f(p,s)f(p,T,s)=0s(8)(A4)T=y

h=f(p,T)(6)(A5)h=y7T=f(p,h)f(p,T,h)=0h1/7

(8)(A5)T=yρ=f(p,T)(6)(A5)ρ=5000-1000yT=f(p,ρ)f(p,T,ρ)=05-ρ/1000(8)(A4)T=yλ=f(p,T)f(p,T,λ)=0100-100λ(6)(A5)λ=1-y/100µ=f(p,T)f(p,T,µ)=

(106µ)1/6

(8)(A5)

µ=106y6

4.AccuracyandcalculationspeedofexplicitfastcalculationformulaforsupercriticalCO2thermodynamicproperties

Inordertoverifythecalculationaccuracyandspeedofthefastcalculationformulae,thecalculationresultsandtimeofthefastcalculationformulaearecomparedwiththoofREFPROP7.1.10,000points,whicharedistributeduniformlyintheapplicationrangesofthedevelopedcorrelationsforsupercriticalthermalproperties,arecalculated,respectively.Table3showsthedeviationsoftheexplicitformulaefromREFPROP7.01.Itcanbefoundthatthetotalmeanrelativedeviationsofthefastcalculationformulaearelessthan1.8%,anddeviationsforabout91.12%datapointswhicharecalculatedbythefastcalculationformulaearelessthan1%anddeviationsforabout91.12%datapointsarelessthan2%.Inaddition,thedatapointswithhigherdeviationsconcentrateinasmallregionnearcriticalpoint.Table3alsoshowsthecalculationspeedcomparisonoftheexplicitformulaeandREFPROP7.01.ItcanbefoundthatthemeancalculationspeedofthefastcalculationformulaeforthermalpropertiesofsupercriticalCO2isabout130timesfasterthanREFPROP7.01.

Intheconditionthatthetemperatureiscalculatedwhenotherthermalpropertiesaregiven,thespeedadvantageofthefastcalculationformulaearemoreobvious.

Table3ComparisonofcalculationaccuracyandspeedbetweenREFPROP7andfastcalculation

formulae

RelativeDeviationCalculationtimec7.3773MPa～12.0MPa

304.15～393.15K304.15～318.15K318.15～393.15KFormulameana(%)maxb

(%)meana(%)maxb(%)meana(%)maxb(%)tNIST(s)tFCF

(s)NISTFCFtts=f(p,T)0.0912.960.3612.960.030.224.460.06667.58T=f(p,s)0.030.460.040.460.030.1621.46

0.058370.00

h=f(p,T)0.326.551.097.000.171.064.520.07857.94T=f(p,h)0.121.030.241.030.100.3718.50.066280.47ρ=f(p,T)0.3727.221.0928.620.230.864.510.06272.70T=f(p,ρ)0.070.430.110.430.060.211.960.05436.31λ=f(p,T)1.8058.966.6958.960.866.665.470.06682.86µ=f(p,T)

1.3224.766.5224.760.313.345.470.0668

2.79

Totalmeand0.5216.552.0216.780.221.618.290.065131.3

aFCF

NISTNIST1

mean(||/)

N

iY

YYEN=−=

∑whereYFCPisthedatacalculatedbythefastcalculationformulae,YNISTis

gottenfromREFPROP7,Nisthenumberofthetestpoints,andN=10000.

bmaxFCFNISTNISTmax(||/)EYYY=−whereYFCPisthedatacalculatedbythefastcalculationformulae,YNISTisgottenfromREFPROP7.

cThecalculationisdoneonaPCwiththeCPUAMDTurion(tm)64X2.Eachpointofthefunctionforsupercriticalthermodynamicpropertyhasbeencalculate

d10000times.d[]1

(1/)MeandeviationmTotalmeanm=∑,wheremisthenum