Simulation andOptimization inProcessEngineering
TheBenefitofMathematicalMethods inApplicationsoftheChemicalIndustry
Editedby
Prof.Dr.MichaelBortz
Dr.NorbertAsprion
Elsevier
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ISBN:978-0-323-85043-8
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1.Predictionandcorrelationofphysicalproperties includingtransportandinterfacialproperties withthePC-SAFTequationofstate JonasMairhoferandJoachimGross
2.Don’tsearch—Solve!Processoptimizationmodeling withIDAES
LorenzT.Biegler,DavidC.Miller,andChineduO.Okoli
3.Thinkingmulticriteria—Ajackknifewhenitcomesto optimization
NorbertAsprionandMichaelBortz
1.Introduction 57
1.1Shortaccountonmulticriteriaoptimization57
2.Processdesign 60
2.1Continuousdesignvariables61
2.2Discretealternatives62
2.3Theimpactofuncertainties64
2.4Extensiontooptimalcontrol68
3.Modeladjustment,modelcomparisonandmodel-based designofexperiments 69
4.Integratedmodelingandenergeticoptimizationof thesteelmakingprocessinelectricarcfurnaces:An industrialapplication
Jesu ´ sD.Herna ´ ndez,LucaOnofri,andSebastianEngell
1.Introduction 77
2.Electricarcfurnaceprocessmodel 79 2.1HybridEAFprocessmodel79
3.Dynamicoptimizationofthemeltingprofiles 87
3.1Problemstatement87
3.2Ageneralformulationofthedynamicoptimization problem88
3.3Formulationofthedynamicoptimizationproblem oftheEAFprocess88
4.Solutionusingcontrolvectorparametrization 89
4.1Numericalsolutionofthemodel89
4.2Terminationconditions91
4.3Modelvalidationandparameterestimation91
4.4Numericalsolutionoftheoptimizationproblem94
4.5Batchtimeconstraint94
5.Resultsanddiscussions 95
5.1Numericalcasestudy95
5.2Resultsfortherealindustrialprocess97
6.Conclusions 98 References 99
5.Solventrecoverybybatchdistillation—Applicationof multivariatesensitivitystudiestohighdimensional multiobjectiveoptimizationproblems
JanC.Schoneberger,DanielStaak,andJurgenRarey
1.Introduction 101
1.1Separationofacetoneandmethanol102
1.2Continuousseparationprocesses102
1.3Batchprocessesforseparation103
2.Problemdefinition 103
2.1Productspecificationsandconstraints103 2.2Descriptionoftheplant103
3.Literaturereview 106
4.Methodology 109
4.1Heuristicsfortheselectionofasuitablemultipurpose plant109
4.2Toolforrunningflowsheetsimulations110
4.3Algorithmsforoptimizingflowsheet simulations110
4.4Toolforrunningmultivariatesensitivity studies111
5.Setupoftheflowsheetsimulation 111 5.1Thermodynamicmodels111 5.2Screeningmodel112 5.3Low-fidelitymodel115 5.4High-fidelitymodel115
6.Results 117 6.1Screeningmodel117 6.2Low-fidelitymodel121 6.3High-fidelitymodel131 6.4Economicevaluation137 7.Summary
6.Modelingandoptimizingdynamicnetworks: Applicationsinprocessengineeringandenergy supply
JanMohring,JochenSchmid,JarosławWlazło, RaoulHeese,ThomasGerlach,ThomasKochenburger, andMichaelBortz
5.Conclusion
7.Theuseofdigitaltwinstoovercomelow-redundancy problemsinprocessdatareconciliation
FilippoBisotti,AndreaGaleazzi,FrancescoGallo, andFlavioManenti
1.Introduction 161
2.Datareconciliation 163
2.1Variableclassification163
2.2Steady-statedatareconciliation(DR)163
2.3Grosserrordetection165
2.4Grosserroreffectandhowtohandle165
2.5Grosserrordetection:Statisticalmethods166
2.6GEstatisticaldetectionalgorithms167
2.7Numericalmethodforlow-redundantsystem167
3.Clevermeanandclevervariance(cmandcv) 168
4.Medianandmad 170
4.1Dynamicdatareconciliation171
4.2Movingtime-windowapproach172
4.3SolutionofDDRwithorthogonalmatrix173
4.4Implementationandtheroleofdigitaltwin175
5.Industrialcasestudy:ItelyumRegenerationamine washingunit 177
5.1Processdescription177
5.2Assumptions179
6.Results 180
6.1Steady-statedatareconciliationresultsdiscussion180
6.2Grosserrordetectionresultsdiscussion186
6.3Dynamicdatareconciliationcasestudy:Aminetank dynamics189
7.Conclusions 195
7.1Steady-statedatareconciliation195
7.2Dynamicdatareconciliation(DDR)196 Acknowledgments 198 References 198
8.Real-timeoptimizationofbatchprocessesvia optimizingfeedbackcontrol
DominiqueBonvin,GregoryFranc ¸ois,GianlucaRizzi, andMichaelAmrhein
1.Introduction 201
2.Representationofbatchprocesses 203
2.1Distinguishingfeatures203
2.2Mathematicalmodels203
2.3Staticviewofabatchprocess204
3.Numericaloptimizationofbatchprocesses 205
3.1Problemformulation:Dynamicoptimization206
3.2Reformulationofadynamicoptimizationproblemasa staticoptimizationproblem206
3.3Batch-to-batchsolution:Staticoptimization207
3.4Effectofplant-modelmismatch208
4.Feedback-basedoptimizationofuncertainbatch processes 209
4.1Offlineactivity:Determinethefeedbackstructure209
4.2Real-timeactivities:Implementfeedbackcontrol211
5.Illustrativeexample:Batchdistillationcolumn 213
5.1Industrialbatchdistillationcolumn213
5.2Processmodel215
5.3Inputparameterizationoftheimpurityfraction216
5.4Controldesignandperformance218
6.Conclusions
9.Oneconomicoperationofswitchablechlor-alkali electrolysisfordemand-sidemanagement KosanRoh,LuisaC.Bree,KarenPerrey,AndreasBulan, andAlexanderMitsos
1.Introduction 226
2.Operationalmodeswitchingofchlor-alkalielectrolysis
3.Mathematicalformulationforoptimalsizingandoperation ofswitchablechlor-alkalielectrolysis
3.1Operationalmodetransition230
3.2Massbalance231 3.3Powerdemand231
3.4Rampingconstraints232 3.5Costfunction233
4.Casestudy 233
4.1Optimaloperationalbehaviorofswitchablechlor-alkali electrolysis234
4.2Comparisonofswitchablechlor-alkalielectrolysis tootherflexibilityoptions234
4.3Simultaneousoptimizationofplantoversizingand operation238
5.Conclusion
10.Optimalexperimentdesignfordynamicprocesses
SatyajeetBhonsale,PhilippeNimmegeers, SimenAkkermans,DriesTelen,IoannaStamati, FilipLogist,andJanF.M.VanImpe
1.Introduction 243
2.Optimalexperimentdesignformodelstructure discrimination 246 2.1OED/SDinpractice248
Contents
3.Optimalexperimentdesignforparameterestimation 251
3.1Computingparametervariance-covariancematrix252
3.2OED/PEasanoptimalcontrolproblem254
3.3OED/PEinpractice257
4.Advanceddevelopmentsinoptimalexperimentdesign 257
4.1Robustoptimalexperimentdesignforparameter estimation257
4.2Multicriterionoptimalexperimentdesign263
5.Conclusions 268 References
11.Characterizationofreactionsandgrowthin automatedcontinuousflowandbioreactor platforms—FromlinearDoEtomodel-based approaches
TilmanBarz,JulianKager,ChristophHerwig, PeterNeubauer,MarianoNicolasCruzBournazou, andFedericoGalvanin
1.Introduction 273
2.Miniaturizedplatformsandapplications 275
2.1Continuous-flowmicroreactorplatformsinsynthetic chemistry275
2.2Bioreactorplatformswithautomaticliquidhandling277
2.3ApplicationsofDoE,self-optimization,andmbOED—A bibliographicalreview283 2.4Summary297
3.Specialaspectsandchallenges 298
3.1Staticvsdynamicexperimentalconditions298
3.2SequentialplanningandupdatinginmbOED301
3.3Parameteridentifiability303
3.4Bayesianstatistics304
3.5Mathematicalmodeling,softwareandalgorithms306
4.Industryview 310
4.1 mbOED software,flexibility,usability,andrequired expertknowledge311
5.Discussionandconclusions 312 References 313
12.Productdevelopmentinamulticriteriacontext
PhilippSuss,GregorFoltin,MelanieHeidgen, DavidHajnal,JorgeDiaz,HergenSchultze, JochenGattermayer,andStefanLehner
1.Introduction 321 2.Modelfitting 322
2.1Generatingthedata:Designofexperiments323
3.Multicriteriaoptimizationanddecision-making 326
4.Approximatingthesetofefficientproductdesigns
7.Application:Designinganexteriorpaintrecipe
13.Dispatchingforbatchchemicalprocessesusing Monte-Carlosimulations—Apracticalapproachto schedulinginoperations
HeinerAckermann,MichaelHelmling,StefanHoeser, NeeleLeith € auser,MiguelA.Romero-Valle, CarlosTellaeche,andChristianWeiß
1.Introduction
2.Proposedsolution
3.Implementation
3.1Importantstepsfortheimplementationofourdecision supporttoolinpractice357 3.2Thefinalapplication359
4.Beyondreal-timeoperativescheduling
4.1Usecase1:Predictionoffutureeventsandplantstates360 4.2Usecase2:What-ifanalysesforplantexpansion/ optimization361
5.Conclusionsandoutlook
14.ApplicationsoftheRTNschedulingmodelinthe chemicalindustry
HectorD.Perez,SatyajithAmaran,ShachitS.Iyer, JohnM.Wassick,andIgnacioE.Grossmann
1.Introduction
2.ReviewofRTNmodel
2.1Discrete-timerepresentation370 2.2Continuous-timerepresentation372
2.3Discrete-timevscontinuous-time376
3.Industry-leddevelopments
Contributors
Numbersinparenthesisindicatethepagesonwhichtheauthors’contributionsbegin.
HeinerAckermann (339),FraunhoferITWM,Kaiserslautern,Germany
SimenAkkermans (243),BioTeC+,DepartmentofChemicalEngineering,Technology CampusGent,KULeuven,Ghent,Belgium
SatyajithAmaran (365),TheDowChemicalCompany,Midland,MI,UnitedStates
MichaelAmrhein (201),OnlineControl,Lausanne,Switzerland
NorbertAsprion (57),BASFSE,Ludwigshafen,Germany
TilmanBarz (273),CenterforEnergy,AITAustrianInstituteofTechnologyGmbH, Vienna,Austria;KIWI-biolab,DepartmentofBiotechnology,Technische UniversitatBerlin,Berlin,Germany
SatyajeetBhonsale (243),BioTeC+,DepartmentofChemicalEngineering,Technology CampusGent,KULeuven,Ghent,Belgium
LorenzT.Biegler (33),DepartmentofChemicalEngineering,CarnegieMellon University,Pittsburgh,PA,UnitedStates
FilippoBisotti (161),PolitecnicodiMilano,CMICDept.“GiulioNatta”,Centrefor SustainableProcessEngineeringResearch(SuPER),Milano,Italy
DominiqueBonvin (201),Laboratoired’Automatique,EPFL,Lausanne,Switzerland
MichaelBortz (57,143),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
MarianoNicolasCruzBournazou (273),KIWI-biolab,DepartmentofBiotechnology, TechnischeUniversitatBerlin,Berlin,Germany;DataHowAG,Dubendorf, Switzerland
LuisaC.Bree (225),ProcessSystemsEngineering(AVT.SVT),RWTHAachen University,Aachen,Germany
AndreasBulan (225),CovestroDeutschlandAG,Leverkusen,Germany
JorgeDiaz (321),BASFSE,Ludwigshafen,Germany
SebastianEngell (77),ProcessDynamicsandOperationsGroup,Departmentof BiochemicalandChemicalEngineering,TUDortmundUniversity,Dortmund, Germany
GregorFoltin (321),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
GregoryFranc ¸ ois (201),HES-SOValais-Wallis,Sion,Switzerland
AndreaGaleazzi (161),PolitecnicodiMilano,CMICDept.“GiulioNatta”,Centrefor SustainableProcessEngineeringResearch(SuPER),Milano,Italy
FrancescoGallo (161),ItelyumRegenerations.r.l,Lodi,Italy
FedericoGalvanin (273),DepartmentofChemicalEngineering,UniversityCollege London(UCL),London,UnitedKingdom
JochenGattermayer (321),BASFSE,Ludwigshafen,Germany
ThomasGerlach (143),BayerAG,BuildingE41,Leverkusen,Germany
JoachimGross (1),InstituteofThermodynamicsandThermalProcessEngineering, UniversityofStuttgart,Stuttgart,Germany
IgnacioE.Grossmann (365),CarnegieMellonUniversity,Pittsburgh,PA,United States
DavidHajnal (321),BASFSE,Ludwigshafen,Germany
RaoulHeese (143),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
MelanieHeidgen (321),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
MichaelHelmling (339),FraunhoferITWM,Kaiserslautern,Germany
Jesu ´ sD.Herna ´ ndez (77),ProcessDynamicsandOperationsGroup,Departmentof BiochemicalandChemicalEngineering,TUDortmundUniversity,Dortmund, Germany;AcciaiSpecialiTerni(AST),Terni,Italy
ChristophHerwig (273),InstituteofChemical,EnvironmentalandBioscience Engineering,TechnischeUniversit€ atWien,Vienna,Austria
StefanHoeser (339),BASFItaliaS.p.A.,PontecchioMarconi,Italy
ShachitS.Iyer (365),TheDowChemicalCompany,Midland,MI,UnitedStates
JulianKager (273),CompetenceCenterCHASEGmbH,Linz,Austria
ThomasKochenburger (143),BayerAG,BuildingE41,Leverkusen,Germany
StefanLehner (321),BASFSE,Ludwigshafen,Germany
NeeleLeith€ auser (339),FraunhoferITWM,Kaiserslautern,Germany
FilipLogist (243),BioTeC+,DepartmentofChemicalEngineering,Technology CampusGent,KULeuven,Ghent,Belgium
JonasMairhofer (1),BASFSE,Ludwigshafen,Germany
FlavioManenti (161),PolitecnicodiMilano,CMICDept.“GiulioNatta”,Centrefor SustainableProcessEngineeringResearch(SuPER),Milano,Italy
DavidC.Miller (33),NationalEnergyTechnologyLaboratory,Pittsburgh,PA,United States
AlexanderMitsos (225),ProcessSystemsEngineering(AVT.SVT),RWTHAachen University,Aachen,Germany;JARA-ENERGY,Aachen,Germany;Instituteof EnergyandClimateResearch:EnergySystemsEngineering(IEK-10), ForschungszentrumJulichGmbH,Julich,Germany
JanMohring (143),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
PeterNeubauer (273),KIWI-biolab,DepartmentofBiotechnology,Technische Universit€ atBerlin,Berlin,Germany
PhilippeNimmegeers (243),BioTeC+,DepartmentofChemicalEngineering, TechnologyCampusGent,KULeuven,Ghent,Belgium
ChineduO.Okoli (33),NationalEnergyTechnologyLaboratory,Pittsburgh,PA, UnitedStates
LucaOnofri (77),AcciaiSpecialiTerni(AST),Terni,Italy
HectorD.Perez (365),CarnegieMellonUniversity,Pittsburgh,PA,UnitedStates
KarenPerrey (225),CovestroDeutschlandAG,Leverkusen,Germany
JurgenRarey (101),RareytecCo.,Ltd,NakhonRatchasima,Thailand
GianlucaRizzi (201),OnlineControl,Lausanne,Switzerland
KosanRoh (225),ProcessSystemsEngineering(AVT.SVT),RWTHAachen University,Aachen,Germany;DepartmentofChemicalEngineeringandApplied Chemistry,ChungnamNationalUniversity,Daejeon,RepublicofKorea
MiguelA.Romero-Valle (339),BASFSE,Ludwigshafen,Germany
JochenSchmid (143),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
JanC.Sch€ oneberger (101),ChemstationsEuropeGmbH,Berlin,Germany
HergenSchultze (321),BASFSE,Ludwigshafen,Germany
DanielStaak (101),LonzaAG,Visp,Switzerland
IoannaStamati (243),BioTeC+,DepartmentofChemicalEngineering,Technology CampusGent,KULeuven,Ghent,Belgium
PhilippSuss (321),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
DriesTelen (243),BioTeC+,DepartmentofChemicalEngineering,Technology CampusGent,KULeuven,Ghent,Belgium
CarlosTellaeche (339),BASFSE,Ludwigshafen,Germany
JanF.M.VanImpe (243),BioTeC+,DepartmentofChemicalEngineering, TechnologyCampusGent,KULeuven,Ghent,Belgium
JohnM.Wassick (365),TheDowChemicalCompany,Midland,MI,UnitedStates
ChristianWeiß (339),FraunhoferITWM,Kaiserslautern,Germany
JarosławWlazło (143),FraunhoferInstituteforIndustrialMathematics(ITWM), Kaiserslautern,Germany
Predictionandcorrelation ofphysicalpropertiesincluding transportandinterfacial propertieswiththePC-SAFT equationofstate
JonasMairhofera andJoachimGrossb
aBASFSE,Ludwigshafen,Germany, bInstituteofThermodynamicsandThermalProcess Engineering,UniversityofStuttgart,Stuttgart,Germany
Thereliabilityofaprocesssimulation,toalargeextent,isdeterminedbythe fitnessoftheappliedphysicalpropertymodel.Adequatemodelsarerequired formanydifferentphysicalproperties:thesimulationofadistillationcolumn usingtheequilibriumstageapproachmayrequireamodelforactivitycoefficients,vaporpressure,enthalpy,andpossiblyvaporphasefugacitycoefficients. Rate-basedprocessmodels,inaddition,needtransportpropertiessuchasviscosity,thermalconductivity,ordiffusioncoefficientsasinput.Inordertodeterminevolumeflowsfrommassormolarflows,forexampleforequipment sizing,thedensityofastreamneedstobeknown.Furthermore,entropyplays animportantroleinthesimulationofcompressorsorpumps,etc.Highdemands areplacedonthephysicalpropertymodels:theyhavetoaccuratelycorrelate availableexperimentaldata,butalsoshowrobustnessinextrapolatingover largerangesoftemperatureandpressure.Theyhavetobeabletomodelthe thermodynamicbehaviorofspeciesandmixturesexhibitingcomplexmolecular interactions.Atthesametime,theycannotbearbitrarilycomplexinorderto ensurereliablenumericalsolutionsandtoavoidprohibitivelylongruntimes ofaprocesssimulation.
Severalalternativesexistforprovidingtherequiredphysicalproperties.One ofthemostwidelyusedapproachesinaprocesssimulationisinapplyingan activitycoefficientmodel,suchasthe nonrandomtwo-liquidmodel (NRTL) [1] formodelingtheliquidphasenonidealityincombinationwithdedicated, usuallyempiricalcorrelationsforthepure-componentvaporpressures,liquid
https://doi.org/10.1016/B978-0-323-85043-8.00002-7
andvaporphaseheatcapacities,liquiddensityandfurtherrequiredproperties. Thegasphaseismostoftenapproximatedasanidealgas.Thisapproach showsseveraldrawbacks:itisnotsuitedforhigh-pressureapplicationsbecause thepressuredependenceoftheactivitycoefficientsisusuallyneglected,and theidealgasapproximationwillnotbejustified.Thenonidealityofthegas phasethenalsohastobeaddressedincalculatingthevaporenthalpy.Furthermore,specialtreatmentisnecessaryforcomponentswithacriticaltemperature lowerthanthesimulationtemperature.
Anequationofstatemodelcomplementedwithexpressionsforidealgas heatcapacitiesofallpurespeciescanprovideallstaticthermodynamicproperties.Furthermore,modelsfortransportcoefficients,suchasviscosity,thermal conductivity,ordiffusioncoefficients,existwhichbuildonoutputsofan equationofstate.However,simplecubicequationsofstatesuchasthePengRobinson [2] orSoave-Redlich-Kwong [3] equationsofstatecanonlybe appliedtosimplemoleculesandingeneral,donotproduceresultsthatareaccurateenoughforallrequiredpropertiesinmostreal-worldprocessessimulation applications.Theneedformoreaccuratethermodynamicmodelsapplicable alsotomoleculesshowingcomplexmolecularinteractionssuchashydrogenbondingorpolarinteractionsledtothedevelopmentofmoresophisticated, physically-basedequationsofstatesuchasthe statisticalassociatingfluidtheory (SAFT)developedbyChapmanetal. [4,5] andJacksonetal. [6] basedon theworkofWertheim [7–10].SAFTleadstoanexpressionfortheresidual Helmholtzenergy Ares(T,ρ,x) ¼ A(T,ρ,x) Aig(T,ρ,x),where Aig denotesthe Helmholtzenergyoftheidealgasattemperature T,numberdensity ρ andmole fractions x.Thevalueof Ares isobtainedasthesumofdifferentcontributions. Eachcontributiontakesintoaccountaspecifictypeofmolecularinteraction. ThedifferentHelmholtzenergycontributionsaredevelopedusingperturbation theorywhich(undersuitableconditions)allowstoobtainthethermodynamic propertiesofatargetfluidwithspecifiedinteractionpotentialfromthepropertiesofareferencefluidwithasimplerinteractionpotentialandknownproperties.TheadvantageofSAFTisthatitallowstoincludeacontributionforhighly directionalattractiveinteractionssuchashydrogen-bondingwhichmakesita suitablechoiceformodelingthepropertiesofmolecules(andtheirmixtures) exhibitingsuchcomplexmolecularinteractions.DifferentSAFT-typeequationsofstatecanbederiveddependingonthechoiceofreferencefluidand interactionpotentialofthetargetfluid.
AdetaileddescriptionofthefundamentalsofSAFTcanbefoundinthebook bySolana [11].AcomprehensivelistofsuccessfulapplicationsofSAFT-type equationsofstatetocomplexsystemsincludingpolarandassociatingmolecules,polymers,ionicliquids,pharmaceuticalsaswellasbio-moleculescan befoundinreviewarticles [12–18].
Thescopeofthischapteristointroducethe perturbed-chainstatisticalassociatingfluidtheory (PC-SAFT).Theequationstoimplementthemodelare givenin Section1.Thedeterminationofmeaningfulpure-componentand binaryinteractionparametersisthetopicof Section2 Section3 presents
group-contributionmethodsforPC-SAFTwhichallowtopredictthemodel parametersinthefrequentlyencounteredsituationthatnotenoughexperimental dataisavailableforagivenmolecule,thuspreventingparameterregression. Finally, Sections4and5 areconcernedwithcorrelatingorpredictingthetransportandinterfacialproperties,respectively.
1ModelequationsofPC-SAFT
InthecaseofPC-SAFTdevelopedbyGrossandSadowski [19–21],the hard-chainfluidisusedasthereference.Eachchainismadeupof m bonded hard-spheresegmentsofdiameterparameter σ .Attractiveinteractionssuch asdispersiveinteractionsorhydrogenbondingbetweenthechainsarethen addedasperturbationstothispurely-repulsivereferencefluid.ThedimensionlessresidualHelmholtzenergy a res ≡ Ares/NkT withmoleculenumber N and Boltzmann’sconstant k isobtainedas ares ¼ ahc + adisp + aassoc + add + aqq + adq (1.1)
TheHelmholtzenergyhasseveralcontributions,namelythehard-chainreferencefluid, a hc,thechangeinHelmholtzenergyduetodispersiveinteractions betweenthechains, adisp,highlydirectionalattractiveinteractionssuchas hydrogen-bonding, a assoc,anddipole-dipole,quadrupole-quadrupoleaswell asdipole-quadrupoleinteractions.Withpolarcontributionsincluded,themodel isusuallyreferredtoas perturbed-chainpolarstatisticalassociatingfluidtheory (PCP-SAFT).Inthefollowingsection,thevarioustermsofEq. (1.1) willbe presentedforamixtureattemperature T,numberdensity ρ,andmolefractionof component i, xi.Usingbasicthermodynamicrelationships,allotherthermodynamicpropertiescanbeobtainedasderivativesof a res . TheHelmholtzenergyofthehard-chainfluidisgivenby
where a hs denotestheHelmholtzenergyofthehard-spherefluid.Theaverage segmentnumberofthemixtureiscalculatedas m ¼ Pi xi mi with mi asthenumberofsegmentsonachainofcomponents i.Wenotethatparameter mi isin SAFTmodelsrelaxedtobeareal-valued(ratherthaninteger-valued) parameter.
Thevalueof a hs isobtainedfromtheaccurateequationofstateforthehardspherefluidpresentedbyBoublı´k [22] andMansoorietal. [23]
withdensitymeasures ζ n definedas
Here, di denotesthetemperature-dependentsegmentdiameterofcomponent i di T ðÞ¼ σ i 1 0:12exp
withthedispersiveinteractionenergy εi/k betweensegmentswithinthechainof component i.Furthermore,theradialdistributionfunctionofthehard-sphere fluidatcontactdistanceiscalculatedas
ThecontributiontotheHelmholtzenergyduetodispersiveinteractions betweenthechainmoleculesisdevelopedasaperturbationtothehard-chain referencefluidusingthesecond-orderperturbationtheoryofBarkerandHenderson [24,25] extendedtochainfluids [19] andaperturbationpotentialof Lennard-Jonestype,with
Thecontributionsoffirstandsecond-orderareobtainedas
andpackingfraction η ¼ ζ 3.Thevaluesfor εij and σ ij aredeterminedfromthe Lorentz-Berthelotcombiningrulesas εij ¼ εi εj p and σ ij ¼ 0.5(σ i + σ j).Asdiscussedin Section2.2,itisoftennecessarytointroduceadjustablebinaryinteractionparameters(BIP)forcalculating εij and σ ij inordertoimprovethe descriptionofmixtureproperties.Theperturbationapproachrequirestheevaluationofintegralsoverthepair-correlationfunctionofthereferencefluidand theperturbationpotential.InPC-SAFT,theseintegralsareapproximatedas power-seriesindensity
withcoefficients ai m ðÞ and bi m ðÞ thatdependsontheaveragesegmentnumberofthemixture.Asimplebutaccurate [19] dependenceonsegment-number wastakeninanalogytoachain-formationtheorybyLiuandHu [26],as
Themodelconstants a0i, a1i, a2i aswellas b0i, b1i,and b2i wereadjustedto experimentalvaporpressureandPvT-dataof n-alkanes.Theirvaluescanbe foundintheoriginalpublication [20].
Inordertoincludehighlydirectionalattractiveinteractionssuchashydrogenbonds,associationsitesareplacedonthechainmolecules.Thesesitescan beofdifferenttypesandonlyinteractionsbetweencertainsitetypesareallowed tooccur.Sitetypescanforexamplerepresentelectrondonorsoracceptors. Interactionsarethenallowedbetweenadonorandanacceptorsitebutnot betweentwodonorortwoacceptorsites.Earlyclassificationofassociation-site schemesforseveralimportantchemicalfamiliescanbefoundintheworkof HuangandRadosz [27].ThefinalexpressionfortheHelmholtzenergycontributionduetoassociationis
where Γi denotesthesetofassociationsiteslocatedonamoleculeofspecies i and χ i A isthefractionofnonbondedassociationsites A onmoleculesoftype i.Thevalueof χ i A hastobedeterminedbysolvingthesetofnonlinearequations givenby
Here, εAi, Bj HB and κ Ai, Bj HB denotetheassociationstrengthbetweensite A on moleculesofcomponent i andsite B onmoleculesofcomponent j andthe dimensionlessvolumeinwhichassociationbetweenthetwositescanoccur, respectively.TheirvaluesareobtainedfromthefollowingcombiningrulessuggestedbyWolbachandSandler [28]
Strictlyspeaking,combiningruleshavenosoundjustificationforcrossassociationsbetweentwo(self-)associatingmolecules i and j (asopposedto dispersiveinteractions,whereapproximationscanbemadetoderivethe Berthelot-Lorentzcombiningrules).Cross-associationscanbedetermined throughquantummechanicalcalculations.Inmanypracticalapplications,however,thesimplecombiningrules,Eqs. (1.18)and(1.19),canproducesuitable approximationsofthecross-associationparameters.Clearly,binaryinteraction parametersmaybeintroducedinEqs. (1.18)and(1.19) toimproveresults,see Section2.2.EfficientschemestosolveEq. (1.16) andtodeterminederivatives of a assoc havebeendevelopedforexamplebyMichelsen [29],Michelsenand Hendriks [30],Tanetal. [31],andLangenbachandEnders [32].
Inmixturesofself-associatingmolecules i withmolecules j thatdonotselfassociatewhich,however,provideprotondonororacceptorsitesthatcanform cross-hydrogenbondswithmoleculesoftype i,itisoftenadvisabletoassigna nonzerovaluefor κ Ai, Bj HB tocomponent j [33].Pure-componentresultsforcomponent j remainunchangedbecausethevaluefor εAi, Bj HB issettozero.However, themodelnowtakesthecross-associationbetweenmolecules i and j into accountbecauseboth,cross-associationvolume, κ Ai, Bj HB ,aswellascrossassociationenergy, εAi, Bj HB ,arenonzero.Tofine-tunemixtureresultsaftera (moreorlessarbitrary)valueforassociationvolumewasassignedtocomponent j,BIPmaybeintroducedaspresentedin Section2.2.
Helmholtzenergycontributionsforpolarmoleculesweredeveloped,among others,byGross [34] forquadrupole-quadrupoleinteractions,GrossandVrabec [35] fordipole-dipoleinteractions,andVrabecandGross [36] fordipolequadrupoleinteractions.Inallthreecases,theresultingexpressionsare obtainedfromathird-orderperturbationapproachpresentedbyStelletal. [37,38] extendedtothetwo-centerLennard-Jonesfluid.ThethreepolarHelmholtzenergycontributionsaresimilarinstructureandonlytheexpressionfor thedipole-dipolecontribution a dd ispresentedhere.Thereaderisreferredtothe originalpublicationsfordetailsonthequadrupole-quadrupoleterm, aqq,andthe dipole-quadrupolecontribution, adq.
Fordipole-dipoleinteractions,theHelmholtzenergyisgivenastheresultof thethird-orderperturbationinPadeapproximation
Thesecondandthird-orderperturbationtermsare
and add 3 ¼ 4
Thedimensionlesssquareddipolemomentisobtainedas
withdipolemomentofcomponent i, μi.Theperturbationapproachrequiresthe evaluationofintegralsoverthepair-correlationandthree-bodycorrelation functionsofthereferencefluid.Theseintegralsareapproximatedassimple powerfunctionsofdimensionlesspackingfraction η:
and
Thecoefficients an,ij, bn,ij,and cn,ijk dependonchainlength m as
Thefollowingcombiningrulesareusedfor mij and mijk
Themodelconstants a0
wereadjustedto theresultsofmolecularsimulationsofStolletal. [39].Theirvaluescanbe foundintheoriginalpublication.Insummary,amoleculeofcomponent i thatismodeledasnonassociatingandnonpolarischaracterizedbythree pure-componentmodelparameters:segmentnumber mi,segmentdiameter parameter σ i,anddispersiveenergy εi.Toincludeassociation,twomoreparametersarerequired:associationstrength εAi, Bj HB andassociationvolume κ Ai, Bj HB .The polarcontributionsrequirethevalueforthedipoleorquadrupolemoment.For both,literaturevaluescanbeused.
