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ChineseJournalofChemicalEngineering,16(1)1115(2008)

    ;GlobalOptimizationfortheSynthesisofIntegratedWaterSystems ;withParticleSwarmOptimizationAlgorithm

    ;LUOYiqing(罗稀青)andYUANXigang(袁希钢)

    ;StateKeyLaboratoryofChemicalEngineering,ChemicalEngineeringResearchCenter,TianjinUniversity,Tianjin

    ;300072,China

    ;AbstractTheproblemofoptimalsynthesisofanintegratedwatersystemisaddressedinthisstudy,wherewater

    ;usinglZIrocessesandwatertreatmentoperationsat”ecombinedintoasinglenetworksuchthatthetotalcostottresh

    ;waterandwastewatertreatmentisgloballyminirmzed.Asuperstructurethatincorporatesallteasibledesignalterna

    ;tlvesforwastewatertreatment,reuseandrecycle,isSVntneSlzedwltnanonllnearprogra

    mmlngmode1.Anevolu-

    ;nonaryapproach--an~mprovedparhcleSWarn1optlrmZationisproposedtoroptlrmzmgsucnsystems.1wos~mple

    ;examplesarepresentedtoillustratetheglobaloptimizationoflntegratedwaternetworksusingtheproposedalgorithm.

    ;Keyworasmtegratedwaternetwork.watermlnlrmzatlon.parhcleswarn1opnm~zanon ;1INTRoDUCTIoN

    ;Waterisoneofthemost1mportantnaturalre

    ;sourcesbeingusedintheprocessindustry.Manyin.

    ;dustrialprocessesneedwateraseitherrawmaterialor

    ;utility,dischargingitintotheenvironmentafteritsuse.

    ;Theextremelyhightoxicityofeffiuentfromtheproc.

    ;essesandtheever?tighteningenvironmentalcon?-

    ;straintsjustifytheneedforappropriateandresponsi

    ;bleutilizationofwaterintheindustry.anddevelop.

    ;mentofmethodologiestominimizewateruseinin

    ;dustrieshasbecomeanimportantareainprocessinte

    ;gration.Numeroussystematicapproachesincluding

    ;conceptua1designmethods【卜5]theapproachinde.

    ;signingwaternetworksbasedonthenecessarycondi

    ;tionsofoptimalityf61andmathematicalmethodsf7101

    ;havebeenpublished.Thesedesignconceptshavebeen

    ;successfullyappliedtoimprovetheefficiencyofthe

    ;watersystemsandreducetheaqueousemissionsto

    ;theenvironment.Inthemathematicalprogramming

    ;approach,thereisaMixedIntegerNon.LinearPro. ;grammingfMINLP)mode1.whichisdecomposedinto ;asequenceofMixedIntegerLinearProgramming ;rMILP)problemstoapproximatetheoptima1solution. ;Bagajewiczeta1.l1]proposedamethodtotransfornl ;theformulationofamulti??contaminantlarge?-scale ;watersystemfromanon.1inearprogram(NLP)toa ;linearprogramfLP1andsolvedittooptimality.The

     ;superstructureoptimizationproblemwasfurtherex

    ;tendedbyLeeandGrossmann,Karuppiahand

    ;Grossmannfl2,l31,whoforrnulatedthedecentralized ;wastewatertreatmentnetworkasanon..convexGen.. ;eralizedDisjunctiveProgramfGDP)andsolvedthe ;problemtoglobaloptimality.

    ;UPtopresent.mostlymethodo1ogiespresentedin ;thepublishedreportonwaterminimizationmainly ;focusontheissueofminimizingwastewatergenera

     ;tioninwaterusingprocesses,separatelyfromthede

    ;signofeffiuenttreatmentsystemsf14,l51.Thereare ;veryfewstudiesontheintegrationofwaterusingand ;solvedtheproblemofoptima1waterallocationina ;petroleumrefinery.Theygeneratedasuperstructure ;allowingforallwaterreuseandregenerationpossi

    ;bilities,andthenmathematicallyoptimizedit.Similar ;studyhasbeendonebyHuangeta1.171,whopre.

    ;sentedatheoreticalmodelforconstructinganoptimal ;waterusageandtreatmentnetworkinachemicalplant. ;Inbothoftheabove.mentionedworks.theintegrated ;networksweremodeledasnonlinearprogramming

    ;problemsandthenoptimized.Globaloptimalityisnot ;guaranteedineitherofthem.KaruppiahandGross. ;mannl81generalizedthesynthesisproblembypro. ;posingasuperstructure,similartothatbyTakamaet ;Z.f161forthedesignofintegratedwatersystemsthat ;combinesthewaterusingandwatertreatingunitsina ;singlenetwork,andproposedanewspatia1branchand ;contractalgorithmforthegloba1optimizationofsuch ;networks.A11theabove.mentionedworks,however, ;assumedthatthewastewatertreatmenttechnologies ;aredescribedbysimplelinearmodelswithconstant ;removalratiosforeachcomponent..Thisisnotalways ;trueinmanytreatmentunits,suchasbidoxidation,

    ;carbonbed,centrifugeseparation,chemicaloxidation, ;crystallization,evaporation,filtrationandsoon,for

    ;whichaconstantremovalratioshouldnotbeassumed. ;Infact.theperformanceofsuchprocesseswillbe ;moreeasilycharacterizedintermsoffixedoutlet ;concentration.Sointhisstudy,toovercomethe ;abovementionedproblems,morerealisticassump

    ;tionsaremaderegardingtheoutletconcentrationsof ;thetreatmentunits.Thesuperstructure.whichincor. ;poratesal1thefeasibledesignalternativesforwater ;treatment.reuseandrecycle,1s1nitiallyforrnulatedas ;anNLPproblem,andanimprovedPSOalgorithmf191 ;forsolvingnon.convexNLP,MINLPproblemswith ;equalityconstraintsisproposedtooptimizetheinte

    ;gratedwatersystems.Twoexampleproblemsaresolved ;todemonstratetheperformanceoftheproposedmethod. ;2PRoBLEMSTATEMENT

    ;treatingprocessesintoasinglesystem.Theseminal ;articleinthisareawasbyTakamaeta1.

    16],whoGivenisanintegratedwaternetworkthatconsists ;Received20070510,accepted20071027.

    ;SupportedbyTianjinMunicipalScienceFoundation(No.07JCZDJC02500)

    ;Towhomcorrespondenceshouldbeaddressed.Email:yuanxg@tju.edu.cn ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.1,February2008 ;ofNwaterusingprocessunitsthatgeneratecontami- ;nantsandMtreatmentunitsthatselectivelyremove ;them.Foreachcontaminantk,theinletandoutlet ;concentrationofeachwaterusingunitJhasanupper ;limitc”andvJ’k,

    ;.

    ;m

    ;u

    ;a

    ;t

    ;x

    ;,

    ;andhasafixedcontaminant

    ;load.Eachwatertreatmentunitthasanupperinlet ;J一一

    ;concentrationlimitC,in”andafixedoutletconcen.

    ;trationoutwhichrepresentsthetreatmentperform- ;anceoftheunitforeachcontaminantk.Furthermore, ;anenvironmentallimitationconcentrationCmustbe ;satisfiedinthedischargepointwhichwillcollectany ;streamcomingfromwaterusingortreatmentunit.

    ;Theproblemisthentosynthesizeanintegratednet. ;workwiththesewaterusingandwatertreatingunits ;tominimizethetotalcost.

    ;Thefollowingassumptionsareadoptedpriorto ;modelingthesystem.

    ;f11Thetotalflowrateofastreamistakentobe ;equaltothatofpurewaterinthatstreamsincethe ;contfibutionsofcontaminants(in10levels)tothe ;flowsarenegligible.

    ;f21Thecostisdeterminedbytheflowsofflesh. ;waterandtheflowsinsidethetreatmentunits.The ;costsofpumpingandpipelineareneglected. ;f31Thenetworkisoperatedunderisothermaland ;isobaricconditions.

    ;2.1Thesuperstructureoftheintegratedwater ;network

    ;AsseeninFig.1.intheinletofwaterusingproc- ;essunitj,fleshwaterFmixeswithreusedwaste- ;waterfromotherwaterusingprocessiandreused ;wastewaterFt

    ;.

    ;jfromwastewatertreatmentunitt.In

    ;fromotheroperationuniti,F

    ;freshwater,

    ;fromtreatmentunitf

    ;theoutletofprocessJ,theflowrateofwastewater ;reusedbyotherwaterusingprocessiisFj ;.

    ;,theflow

    ;rateofwastewaterdischargedisFj,

    ;.ut,andtheflow

    ;rateofwastewatertreatedbytreatmentunittisFj ;,

    ;f.

    ;Intheinletofwastewatertreatmentunitt(Fig.2), ;wastewaterF|tfromwaterusingunitJmixeswith ;wastewaterF/,f

    ;fromotherwatertreatmentunit1.In

    ;theoutletofprocesst,theflowrateofwastewater ;reusedbywaterusingprocessJistheflowrate ;ofwastewaterdischargedis-0ut,andtheflowrateof ;wastewatertreatedbytreatmentunitlisThe ;wastewaterdischargedfromdischargedpointmust ;satisfyanenvironmentlimitconcentration(Fig.3).

    ;2.2TheNLPmodeloftheintegratedwaternet- ;work

    ;Themathematicalmodeloftheintegratedwater ;networkisbasedupontheoverallandspeciesmass ;balancesaroundeachunitoperation. ;Objectivefunction:

    ;MinJ?+?f?,+ZF/,]1(1)IJt\Jl/l

    ;Where,andarethecostparametersrelated ;relativelytothefleshwaterconsumptionandthewa. ;tertreatmenfiRMBYuankg_..h_..

    ;Theconstraintsequationsaregivenas: ;Totalmassbalanceforwaterusingprocessfand ;treatmentunitt:一一.

    ;

    ;?,,

    ;tootheroperationuniti,,

    ;+

    ;waterusing,.todischa:

    ;processjF

    ;{toothert

    ;Figure1Thesuperstructuremodelofwaterusingnetwork

    ;…………’,?l

    ;treatmentunit,Ft.,

    ;:ltreatmentunitttodiscltoothe~ ;entunittF

    ;operationunit,Fi

    ;argedpoint,n

    ;treatmentunit,

    ;Figure2Thesuperstructuremodelofwastewatertreatmentnetwork

    ;fromoperationunit,.uI

    ;fromtreatmentunit,

    ;.uc

    ;satisfied

    ;t,c

    ;Figure3Thesuperstructuremodelofdischargedpoint ;,

    ;.

    ;:0

    ;(2)

    ;?

    ;

    ;?,+

    ;?

    ;+

;

    ;Chin.J.Chem.Eng.,Vo1.16,No.1,February2008 ;?,()+?,()

    ;c+k=0(3)

    ;?,+?,=??+?,+(4)

    ;Jll

    ;Partialmassbalance.forthecontaminant:atthe ;inletoftheunitoperationjandf:

    ;?,+?,=

    ;f?,+?,,+1(5),lti

    ;?,k+?,Ck=f?,+?,,I.(6)

    ;3l\ilj

    ;concentrationlimitations:

    ;ck

    ;

    ;?k

    ;C”k?Ck

    ;.

    ;(7)

    ;k

    ;,out

    ;?

    ;l,,u

    ;k,ma

    ;Outletconstraints:

    ;?c+?Ck??k(8)

    ;3GLoBALonMIoNALGoRITHMFoR

    ;NoN.CoNVEXNLPPRoBLEMS

    ;Sofar,manymethodshavebeenproposedfor ;solvingNLP/MINLPproblems.suchasbranchand ;boundtechnique,outer-approximation(OA)/equality ;relaxationalgorithm,adaptiverandomsearchmethod,

    ;branchandreducealgorithm.MINLPSimplexSimu

    ;latedAnnealingAlgorithm(MSIMPSA1,andgenetic

    ;algorithmandevolutionstrategiesandsoon.Amongst ;thesemethods,deterministicalgorithmsassumecon

    ;vexitytoinsureconvergencetoglobaloptimumsolu

    ;tion.Astostochasticoptimizationalgorithmsbased ;onadaptiverandomsearch,amaindifficultytopre

    ;ventastochasticmethodfrombeingeffectiveisits ;weaknessinhandinghighlyconstrainedproblems. ;ParticleSwarmOptimization(PSO1methodisa ;populationbasedstochasticoptimizationtechnique ;developedbyKennedyandEberhart20].PSOshares

    ;manysimilaritieswithevolutionarycomputation ;techniques,suchasGeneticAlgorithmsfGA1.How

    ;ever,anessentialfeatureofoptimizationalgorithms ;applicabletochemicalengineeringproblemsisitsca

    ;pabilityofdealingwithnonlinearequalityconstrains. ;Morerecently,animprovedPS0algorithmf191 ;hasbeenproposedforsolvingnonconvexNLPLP

    ;problemswithequalityconstraints.Throughvariables ;partitioningandtearingequalityconstraints.NLPJlP

    ;problemwithnonlinearequalityconstrainscanbe ;transformedintothatwithnoequalityconstraints. ;Mathematically.MINLPproblemisgivenby: ;f1)Minimizef(x,,,)r9)

    ;s.tg(x,Y)?0(10)

    ;h(x,)=0

    ;x?X.Y?Y

    ;f11)

    ;(12)

    ;where,vectorxconsistsof,zcontinuousrealvariables. ;and’,consistsofmintegervariables.cRand

    ;,cNarethesolutionspaceofxandY.respec

    ;tively.f,Y)isanobjectivefunction,g(x,)?0and

    ;h)=0representvectorsofinequalityandequality ;constraintswithdimensionsofPand,respectively. ;IntheimprovedPSOmethod,thecontinuousand ;integervariablesaredecomposedas:

    ;x=[,],Y=[@(13)

    ;Variables?-三『cRandCOEQcN”arede—

    ;finedastheindependentvariablesoftheequalitysys

    ;temgivenbyEq.(11),andthenthesumofnumbersof ;thisvariablesV+Ushouldbethefreedomdegreeofthe ;system.Sovariableandcoaresodefinedthatthe ;followingequalityholds

    ;V+/4=,z+mq

    ;Assumingthatdependentvariablesandcan ;berepresentedas=(,co)and=(,co)

    ;throughEq.(11),thefunctionsinproblem(1)canbe ;representedbytheindependentvariables: ;f(x,Y)=f(v,,co,)

    ;=

    ;f(v,(,co),co,(,co))F(v,

    ;g(x,Y)=g(,,co,)

    ;=g(,(,co),co,(,co))G(v,co)

    ;andtheoriginalproblem(1)canbetransformedt0:

;(2)MinimizeF(V,(14)

    ;s.t.G(v,co;I<0(15)

    ;?占nV,coGQnV(16)

    ;withV={(,:h(v,,)=0forsome?

    NnY}. ;nXand?

    ;Problem(2)isaMINLPproblemwithexclu

    ;sivelyequalityconstraints.ThenaP0Salgorithmcan ;beefficientlyappliedtosolvetheMINLPproblem ;withtheinequalityconstraintshandledbypenalty ;termintheobjectivefunction.Withinthealgorithm. ;thereducedvariablescanbesolveddirectlythrough ;theequalityconstraints.Ifthereducedvariablescan

    ;notbeeasilysolved,multiobJectivestrategycouldbe

    ;usedtohandletheequalityconstraints,byconsidering ;theequalityconstraintresidualsasasubcriterionto

    ;beminimized.Thatis.thesolutionsatisfyingcon

    ;strainsarefirstfoundandtheobiectivefunctionis ;thenevaluated.

    ;Inthisarticle,thisalgorithmisappliedtofindthe ;globaloptimumfortheintegratednetworks. ;4EXAMPLES

    ;Example1:Asafirstexample.anetworkwhich ;has4waterusingprocessunitand2treatmentunit ;involvingonecontaminantisconsidered.Datafor ;waterusi.ngprocessunitandtreatmentunitaregiven ;inTables1and2,respectively.Theenvironmental ;

    ;14Chin.J.Chem.Eng.,Vo1.16,No.1,February2008 ;Table1WaterusingunitdataforExample1 ;Table2Treatmentunitdata

    ;dischargelimitforthecontaminantistakentobe1xl0.

    ;Thecostparameterisassumedtobe0.01RMB ;Yuan?17andforeachofthetwowastewatertreat

    ;mentunitsare0.003RMBYuan.kgand0.O11RMB

    ;Yuan?kgrespectively.Table3showstheoptimal ;networkstructurewithanobiectivevalueof1011 ;RMBYuan.h.

    ;Table3ResultofoptimizationforExample1 ;Unit

    ;Flowrate~10/kg.h

    ;RecyclewaterFreshwaterDischargedwater ;Todepicttheadvantageofoptimalintegrationof ;thewaterusingandwatertreatmentnetworks,thewa

    ;terusingnetworkandthewatertreatmentnetwork

    ;wereoptimizedseparatelyandsequentially.The ;freshwaterisfirstminimizedbyoptimizingthewater ;usingunitnetworkwithoutconsideringwatertreat

    ;ment,andthentheeffiuentwastewaterisintroduced ;tOthetreatmentunitnetwork.whichisoptimizedby ;minimizingitscost.WiththesamedatainTableland ;thesamecostparametersforthesimultaneoussolution, ;theoptimalstructureofwaterusingnetworkwithout ;consideringthetreatmentisgiveninTable4.andthe ;optimalstructureforthewatertreatmentnetworkis ;giveninTable5.Thefreshwaterconsumptionforthe ;waterusingnetworkisfoundtobe90t.hwiththe ;correspondingcostof900RMBYuan.h1.total

    ;wastewatertreatmentcostis985.19RMBYuan.h. ;andthenthetotalcostof

    ;networksis1885.19RMB

    ;theentirenonintegrated

    ;Yuan.h_..Itcanbeseen

    ;thateitherthetotalcostorthefreshwaterusageare ;substantiallygreaterthanthosefortheintegratednet

    ;Table4Theoptimumsolutionforwaterusingnetwork ;Table5Theoptimumsolutionforwastewater ;treatmentnetwork

    ;work(1011RMBYuan.hand20t.h1,respectively)

    ;Thisissimplybecausetheintegrationofthetwonet

    ;worksincreaseseffectivelythepossibilityofwater ;reusenotonlyfromthewaterusingunitsbutalsofrom ;thetreatmentunits.

    ;Example2:Itisa2waterusingprocessunitand ;2treatmentunitsysteminvolvingtwocontaminantsA ;andB,forwhichwaterusingprocessunitandtreat

    ;mentunitdataaregiven

    ;tively.Theenvironmental

    ;contaminants(AandB)is

    ;inTables6and7,respec

    ;dischargelimitforboththe

    ;takentobe1×l0.Table8

    ;Table6WaterusingunitdataforExample2

    ;Table7TreatmentunitdataforExample2

    ;Table8ResultofoptimizationforExample2 ;Unit

    ;Flowrate~10/kg.h

    ;RecyclewaterFreshwaterDischargedwater ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.1,February200815 ;showstheoptimalnetworkstructurewithallobjective ;valueof914.11RMBY_uan.h.

    ;Here,theproblemissolvedseparatelyastwo

     ;non--integratednetworksforwaterusingandwaste--;watertreatment.Theoptimalstructuresforthesepa. ;ratednetworksaregivenby6les9and10.respec. ;tively.Theoptimalfreshwaterconsumptionisfound ;50t-hwithacorrespondingcostof500RMB

    ;Yuan.h.andtheoptimaltreatmentcostis581.62 ;RMBY_uan.h.Andthetotalcostisthen1081.62 ;RMBY_uan?h.Theseareagainsubstantiallygreater ;thallthosefortheintegratedsolution.Therefore.the ;significanceoftheintegrationofthetwonetworksis ;clearlyjustified.

    ;Table9Theoptimumsolutionforwaterusingnetwork ;Table10Theoptimumsolutionforwastewater ;treatmentnetwork

    ;Treatpmcessm

    ;

    ;flow

    ;lh

    ;Dischargedflow

    ;ratex10/kg?h

    ;5CoNCLUSIoNS

    ;Thisarticlepresentsasuperstructurebased

    ;methodforoptimizationofanintegratedwatersystem ;consistingofwaterusingandwatertreatingunits,and ;thismethodandthesignificanceoftheintegrationof ;thetwokindsofnetworksareillustratedwithtwo ;examples.Theintegratedsystemisformulatedasa ;NLPproblem.AnimprovedPSOalgorithm.initially ;developedforsolvingnonconvexNLPl/MINLPprob

    ;lemwithequalityand/orinequalityconstraints,isin

    ;troducedtosolvetheglobaloptimizationproblemof ;suchnetworks.

    ;Itshouldbenotedthatthevalidity?

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