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Flexibility analysis of a distillation column: Indexes
comparison and economic assessment
Alessandro Di Pretoro, Ludovic Montastruc, Flavio Manenti, Xavier Joulia
To cite this version:
Alessandro Di Pretoro, Ludovic Montastruc, Flavio Manenti, Xavier Joulia. Flexibility analysis of a
distillation column: Indexes comparison and economic assessment. Computers & Chemical
Engineer-ing, Elsevier, 2019, 124, pp.93-108. �10.1016/j.compchemeng.2019.02.004�. �hal-02063264�
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This is an author’s version published in: http://oatao.univ-toulouse.fr/23313
To cite this version:
Di Pretoro, Alessandro
and Montastruc, Ludovic
and Manenti, Flavio and Joulia,
Xavier
Flexibility analysis of a distillation column: Indexes comparison and economic
assessment. (2019) Computers & Chemical Engineering, 124. 93-108. ISSN 0098-1354
Official URL :
https://doi.org/10.1016/j.compchemeng.2019.02.004
Flexibility
analysis
of
a
distillation
column:
Indexes
comparison
and
economic
assessment
Alessandro
Di
Pretoro
a,b,
Ludovic
Montastruc
a,∗,
Flavio
Manenti
b,
Xavier
Joulia
aa Laboratoire de Génie Chimique, Université de Toulouse, CNRS/INP/UPS, Toulouse, France
b Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica Giulio Natta, Piazza Leonardo da Vinci 32, Milano 20133, Italy
Keywords: Flexibility Index Distillation Design Economics
Theworldwideshareddefinitionof“optimaldesign” referstothecheapestandsimplestdesignableto performtherequiredjob;mostofthetimethisdefinitionisstrictlyrelatedtogivenoperatingconditions, i.e.theinputvariablesareseldomsubjectedtoconsiderablevariations.However,inprocessengineering, plentyofcasesdon’tfitthisdefinitionduetotheuncertainnatureofthefeedstockneededtobe pro-cessed.Therefore,ifasystemislikelytoundergoseveralandsubstantialperturbations,anapriori flexi-bilityassessmentcanbecrucialforthegoodperformanceoftheoperation.Inchemicalengineeringthe leadingseparationprocessisdistillation.Hencethefirstpurposeofthispaperistodefineaprocedure andcomparethedifferentflexibilityindexesfoundinliteratureinordertoperformasimpledistillation column flexibilityassessment.The secondgoalofthispaperis tocouplethe flexibilityand economic aspectsrelatedtothedistillation columninvestmentcostsandagaintocomparethedifferentindexes economicbehaviours.
1. Introduction:Ameasureofflexibility
Flexibilityanalysisisastepofprocessdesignprocedurethatis oftenskipped.Sometimes asensitivityanalysisisperformedwith features similar to the flexibility one but in general they don’t overlap.
The standard procedure for chemical plants design consists of the assessment of the optimal design according to the eco-nomic and operational aspects. Nevertheless this optimal design is strictly related to the operating conditions, i.e. perturbations, when present, can seriously turn the tables. In these cases an a priori flexibilityanalysis could beof criticalimportance toassess in which rangeof operating conditions a systemdesign is effec-tivelybetterperformantthananotherone.
Thewordflexibilitycommonlyreferstotheabilitytochangein ordertocopewithvariablecircumstancesbothinapassiveandan activesense;tobemoredetailed,inprocessengineering,flexibility canbedefinedastheabilityofaprocesstoaccommodateasetof uncertainparameters(HochandEliceche,1996).Thisconceptlooks easily understandable butactually the definition standalone says nothingabouthow wecan dealwithit. Thuswe needan
opera-∗ Corresponding author.
E-mail address: ludovic.montastruc@ensiacet.fr (L. Montastruc).
tionaldefinitionand toknow whatflexibility means through the wayitcouldbemeasured.
Theveryfirstpublicationaboutthedefinitionofaflexibility in-dexis provided by Swaney andGrossmann (1985a) andsoon af-ter Sabooetal.(1985)proposedanotherpossibleindextoquantify theflexibilityofaprocessdesign(calledinthiscase“resiliency”). Moreoverthereexistsanadditionalpaperbyboth Grossmannand Morari(1983) wheretheflexibility relatedproblemsandsolution arethoroughlyanalyzedwithapioneeringmethodology.
TheSwaney and Grossmann flexibility index(FSG)
mathemati-callystatesasfollow: Giventhat:
•
θ
N:nominalvaluesofuncertainparameters(basepoint); •θ
+,θ
−:expecteddeviationsforeachparameter;• d:designvariablesassociatedtotheequipmentcapacities;
• z:controlvariables.
Theflexibilityindex,foragivendesignd,isthesolutionofthe problem:
FSG=max
δ
(1.1)s.t.max
θT(δ)minz maxjJ fj
(
d,z,θ
)
≤ 0 (1.2)IfF=1,thedeviations
θ
+ andθ
− canbeaccommodated.In ordertohaveaclearerideaofwhatthesestatementmeanswecan helpourselveswiththeplotin Fig.1.https://doi.org/10.1016/
Fig. 1. Feasibility domain - F SG ( Swaney and Grossmann, 1985a ).
The curve
(d,
θ
) is the so called feasibility domain defined bythe constraints.Constraints can be physical,legal, operational, economic etc. andin general a deeper studyabout theway this regionisoutlinedandtheconditionstobesatisfied,aswellasthe degreesoffreedomanalysis,isneeded.Thisstepprecedesthe flex-ibilityanalysissincethedomainshapeisstrictlyrelatedtothe na-tureofthecasestudy.Asshownbytheoptimizationproblem,theflexibilityindex de-finedasabove(1.1, 1.2),isthemaximumfractionoftheexpected deviation
δ
that can be accommodatedby the system; it graph-icallyrepresents the minimumamong themaximum fractions of thehyperrectangle sides’ lengthsthat is bounded by thefeasible zone.Moreover,forconstraintsjointlyquasiconvexinzand1Dquasi convexin
θ
theproblemcan bedecomposedintotwo level opti-mizationproblem: FSG=min kδ
k (1.3)δ
k=max zδ
(1.4) fj(
d,z,θ
k)
≤ 0 jJ (1.5)
θ
k=θ
N+δ
·θ
k (1.6)andthesolutionliesatavertexofthehyperrectangleallowingus tosolve theoptimizationproblemby evaluating thefeasibility of the design at each vertex. In this way, it can be noted that the explicitsolutionofthemin-maxproblemcanbecircumvented;on theother handcertaintypesofnon-convexdomains mayleadto nonvertexsolutions(GrossmannandFloudas,1987).
Before the introduction of the flexibility study, to move from feasibilitystudytowardsthe designphase,we justneedto know whetherthe projectisfeasible underthenominaloperating con-ditionsornot,butoncewedealwithflexibilitythequalitative an-sweryes/no isnot sufficientanymore. Weneed toquantify“how much” theprojectisfeasibleandtheindependentvariablesranges enclosingthepossibleoperatingconditions.
Asanticipatedatthebeginning,thesameyearMorarietal. pro-poseda “resiliency index” defined as the capability to easily re-coveroradjustto misfortuneorchange, that ismoreor lessthe passive alter ego of the capability to adapt to new, different or changingrequirements,i.e.flexibility.
Fig. 2. Feasibility domain - F SG vs RI ( Saboo et al., 1985 ).
Thisindexisbased onthesame premisesoftheFSG,i.e.even
inthiscasethevery firststep toperformis thedefinitionofthe feasibility domain.Then theresiliency index RIis defined asthe largesttotaldisturbanceload, independentofthedirectionofthe disturbance,a system isable to withstandwithout becoming in-feasible.
Itmathematicallystandsas:
RI=min i
|
lmax i|
(1.7) s.t.{
max j fj(
θ
)
≤ 0,∀
l:|
li|
i ≤ RI}
(1.8)thiscorrespondsto inscribingthelargestpossiblepolytope inside thefeasible regiondefinedby theinequalities hereabove.The RI isthen equalto thedistancebetweena vertexVofthe polytope andthenominaloperatingpoint0asshownin Fig.2.
ThemainadvantageoftheRIcomparedtotheFSGisthatit
re-quiresalowercomputationaleffort;thiscanbeeasilyfigured out inthe caseofan n-dimensionalproblemwhere avertexanalysis hasto beperformed:in thehyperrectanglecasewe have2n
ver-tices to calculatewhile in the polytope case we have just2n of them.
Moreover, the disturbance region measured in the first case mightbepracticallymoreinteresting becauseitexpressesdirectly the disturbance load allowed in the directionof each parameter independentlyontheothers.
Forthesamereason,sincethehyperrectanglessideshavetobe parallel to the axes even ifthis configuration doesn’t allow the biggestpossiblerectangle,theFSG indexresultstobe very
conser-vativeandsignificantly underestimate theactual flexibility ofthe system.
Thisrepresentationofthefeasibleandexpectedpossiblezones doesnotreflectthoroughlytherealworldsincenotallthepossible operatingconditionsareequallyprobable;hencethiskindof anal-ysis results in rather conservative estimates. In many real world applications,however,dataare usuallyavailable thatallow a bet-terdefinitionofuncertaintyinastatisticalsense.
For this reason a novel flexibility analysis approach for pro-cesses withstochastic parameters was thenproposed in1990 by Pistikopoulos and Mazzuchi (1990). It’ s shown that the flexibil-ityindexcancorrespondtoamultivariate cumulativedistribution functiontransformingtheoriginalconstraintspacetothespaceof stochasticallydependentflexibilityfunctionbymeanofthe analyt-icalpropertiesoftheflexibilityproblem.
Fig. 3. Feasibility domain - F SG vs SF ( Pistikopoulos and Mazzuchi, 1990 ).
Giventhefeasibilityregionconstraintsinequalitiesas:
(
d,θ
)
≤ 0 (1.9)theequalitydefinestheboundaryofthefeasiblezone. ThestochasticflexibilityindexSFcanbedefinedas:
SF =Pr
{
(
d,θ
)
≤ 0}
= {θ:(d,θ )≤0} ... PD(
θ
)
dθ
(1.10)wherePDisthejointdistributionfunctionoftheuncertain
param-eters
θ
.ThecomparisonbetweenthestochasticflexibilityindexSF and theSwaneyandGrossmannisbettershownin Fig.3.
Withthismethodologywecancalculateaweightedestimation offlexibility.Thus,ifsomethingisunlikelytohappenandour sys-temcannotwithstandthisoperatingconditions,itonlyslightly af-fects thefinal value of theflexibility index providinga measure-mentlessconservativeandmoreadherenttoreality.
Obviously,theothersideofthecoinisthestrictdependencyof thisindexon theavailabilityof probabilitydistribution data,that isifnodataareavailablethismethodologyisclearlyunusable.
Anadditionalproblemtobesolvedwaspointedoutin1995by Dimitriadis andPistikopoulos(1995)andit consistsinthe evalu-ation of flexibility takinginto account the dynamicsof the stud-iedsystem. Thistopicwasalreadypointedoutby Grossmann and Morari (1983)few yearsbefore butPistikopoulosisthe veryfirst onetodefineaspecificindexnamelyDynamicFlexibility(DF)that takesinto accountthe evolutionof thesystem. Actuallythe defi-nition itselfof DFisnot so differentsince it isa modification of thepreviousindexFSG.However, itsintroductionallowsthestudy
ofasystemtakingintoaccountitscontrolloopsandtheirtuning, thereforefrom anoperational point ofview its introductionisof criticalimportance.
The Dynamic Flexibilityproblemis introduced herebelow for literature coverage reasons. Nevertheless this index will not be takenintoaccount intheflexibilityanalysisproposed inthenext sectionboth becausenootherdynamicflexibilityindexesto com-pareitwithhavebeenfoundinliteratureandbecausetheanalysis referstosteadystateconditions.
The definition of the Dynamic Flexibility follows the path of the FSG considering theuncertain andcontrolparameters namely
θ
andzasafunctionoftime,thereforethedynamicflexibility in-dexevaluationproblembecomes:DF
(
d)
=maxδ
(1.11) s.t.χ
(
d)
= max θ(t)T(δ,t)z(mint)Z(t)jJmax,t[0,H]fj(
d,x(
t)
,z(
t)
,θ
(
t)
,t)
≤ 0 (1.12) s.t.h(
d,x(
t)
,z(
t)
,θ
(
t)
,t)
=0 (1.13) x(
0)
=x0 (1.14)δ
≥ 0 (1.15) T(
δ
,t)
={
θ
(
t)
|
θ
N(
t)
−δ
·θ
−(
t)
≤θ
(
t)
≤θ
N(
t)
+δ
·θ
+(
t)
}
(1.16) Z(
t)
={
z(
t)
|
zL(
t)
≤ z(
t)
≤ zU(
t)
}
(1.17)Qualitatively, the dynamic flexibility index, DF, represents the largestscaleddeviationoftheuncertainparameterprofilethatthe design can tolerate while remaining feasible within the horizon considered.
The dynamic flexibility index problem is a two-stage, semi-infinite, dynamic optimization problem with an infinite number of decision variables. Therefore, in order to solve it, an ad hoc methodology to reduce the dimension of the problemhas to be implemented.
Finally,the most recent flexibility indexfound inliterature is provided by Lai and Hui (2007, 2008). It has the aim to over-come the problems related to previous indexes, i.e. the require-mentofnominalpointandtheconsiderationofthecritical uncer-taintyonly(causinganunderestimation)forFSG andRI,aswellas
theavailability ofthe probability distributionofall uncertain pa-rametersatthedesignstage fortheSF.Thisnewflexibilitymetric is much easierto use anddoes not need a lot ofcomputational effortoravailabledataanditisdefinedasfollow.
LetV0 bethevolumedefinedbytheuncertainparameters:
V0=
N i=1
(
θ
iU−θ
iL)
(1.18)andVfthefeasiblevolumedefinedastheintersectionofthe
con-strainedvolumeandV0.
ThentheflexibilityindexFV isdefinedastheratiobetweenthe
feasiblespaceandtheuncertainspace:
FV = Vf V0
(1.19)
However,Sfisusuallyirregularinshapeanditsvolume(Vf)
de-termination is not straightforward. To estimate Vf, a constructed
space(Se,theregion outlined bythethick solid line),whose
vol-umedeterminationislessdifficult,canbeinscribedinsideSf.With
acarefulselection oftheshape ofSe,itsvolume(Ve) canbeused
asacloseestimateofVf.
As illustrated in Fig. 4, Se can be constructed by first picking
areferencepoint(PR),whichisnotnecessarilythenominalpoint,
within Sf.Auxiliary vectorsvj withselected directionscan be ra-diatedfromPR.The interception points(Pj)of vj andthe feasible
spaceboundary are obtained. The Se can thenbe constructedby
joiningthesePj pointsaccordingtotheirpositionsinspace.Since differentSecanbegeneratedbydifferentauxiliaryvectordirection
selection schemes, estimation accuracy of Vf andFV will depend
ontheselectionschemeemployed.Thegeneralformulationforthe auxiliaryvectors’positionsina3Dspaceisasfollows:
max xj,yj,zj
Ve
(
xj,yj,zj)
(1.20) fk(
θ
i j)
≤ 0and0≤ xj,yj,zj≤ 1 (1.21)Fig. 4. Feasibility domain - F V ( Lai and Hui, 2008 ). s.t.
θ
i, j=v
i jiji j+
θ
iR,i j=
+·
θ
i(
v
i j≥ 0)
|
−·
θ
i|
(
v
i j<0)
i=x,y,z (1.22)However, whether the feasibility domain is well defined and the constraints equations are know, there’s no need to approxi-mateanymore sincewe canget theexactvalue ofthe Sfvolume
througha multipleintegral atthecostof ahighercomputational effort.
Anadditionalremarkisworthtobedone:thevolumetric flex-ibility index takes into account no perturbation outside the un-certainspaceandtreatevery pointasequallypossible.Therefore, givenitsdefinition,wecanwriteitas:
Fv= Vf V0 = 1 V0 Vf d
θ
= Vf 1 V0 dθ
(1.23)that is the same definition as the stochastic flexibility index for aprobability stepfunction.Apractical applicationofthisanalogy betweenSFandFVwillbeshowninthefollowingchapters.
Finally, on one hand we can say that several flexibility in-dexes have been found in literature but, on the other hand, except the steady state debutanizer case study proposed by Hoch et al. (1995) (FSG), no case studies about distillation have
beenprovided(cf. Table1).Mostofthesystemsfoundinliterature subjectedtoflexibilityanalysishavelinearconstraintsoratleasta quasiconvexfeasibilitydomain;whetherahighercomplexitycan be detected, the flexibility analysishas been conductedwiththe simpleFSG that,duetoitsstraightforwardapplication,resultstobe
theindexusedinthevastmajoritycases.However,thebehaviour of less used and more complex indexes is worth to be studied in deeper and for a wider range of systems as well; therefore, a complete flexibility analysis of a distillation column, inspired by the aforementioned debutanizer example (Hoch et al., 1995), is proposed here below with all the four steady state indexes. An accurate results comparison and economic assessment will follow.
2. Thedebutanizercasestudy
Inordertobetterunderstandthedistillationrelatedapplication of theseindexes,the simple debutanizer casestudyproposed by Hochet al.(1995),withfew modifications, has beenanalyzed in detail.
Inshort,the systemismade upof astandard distillation col-umn, i.e. total condenser, no intermediate feeds/withdrawalsand partialreboiler(Fig.5).
The feedstreamisdefinedby thecomposition andconditions shownin Table2 whilethegivendesignparameters are listedin Table3.
Finallytheuncertainparameters are reportedin Table4, their expectedvariationranges
θ
k± arethesameineitherpositiveornegativedirectionandequalto10%oftheirnominalvalueas sug-gestedinthe paper.Thefeed variationsare relatedto thenature oftheupstream process,theperformancesoftheheatexchangers tothetubesfouling,thewatertemperaturetotheseasonalityand finallytheflooding andweepingvelocityarefunctionofthetrays technologyandstatus.
The specifications are given by the paper as three inequality constraints,namely:
• Maximum molar fractionof butane in the bottom product = 0.01786;
• Maximummolarfractionofpentaneinthedistillate=0.025;
• Minimumpentanerecoveryinthebottom=0.97.
In this paper the two most restrictive equality relationships havebeenselectedtofulfillthetworemainingdegreesoffreedom:
• Molarfractionofbutaneinthebottomproduct=0.01786;
• Pentanerecoveryinthebottom=0.97.
Thecontrolledvariablesarerespectivelytherefluxanddistillate flowrates.
3. Flexibilityanalysis
Flexibilityanalysis can be performedeither duringthe design phaseorforan alreadyexistingequipment(orplant).Inthenext
Table 1
Flexibility studies in literature.
Case study Authors Index
Pump and pipe Grossmann and Morari (1983) FSG
Swaney and Grossmann (1985a) FSG
Floudas et al. (2001) FSG
Lai and Hui (2008) FV
Refrigeration cycle Swaney and Grossmann (1985b) FSG
Reactor-recycle Swaney and Grossmann (1985b) FSG
Heat exchanger network Grossmann and Morari (1983) FSG
Swaney and Grossmann (1985b) FSG
Saboo et al. (1985) RI
Pistikopoulos and Mazzuchi (1990) SF
Dimitriadis and Pistikopoulos (1995) DF
Floudas et al. (2001) FSG
Lai and Hui (2008) FV
Storage tank dynamic system Dimitriadis and Pistikopoulos (1995) DF
Wu and Chang (2017) DF
Debutanizer Hoch et al. (1995) FSG
Hoch and Eliceche (1996) FSG
Reactor + cooler Floudas et al. (2001) FSG
Chemical process with recycle Floudas et al. (2001) FSG
Solar-driven membrane distillation desalination (SMDD) process Wu and Chang (2017) DF
Benzene chlorination reaction system Huang (2017) DF
Batch reactor system Huang (2017) DF
Fig. 5. Debutanizer column layout.
Table 2
Feed conditions and composition. Partial molar flowrates Value Unit
Propylene 0.055 mol/s Propane 0.053 mol/s Butane (lk) 6.863 mol/s Pentane (hk) 2.743 mol/s Temperature Bubble K Pressure 15 ·10 5 Pa Table 3 Design parameters. Desing variables
d Symbol Value Unit
Rectification stages Nr 9 1
Stripping stages Ns 10 1
Column diameter Dcol 0.634 m
Condenser area Acond 40.00 m 2
Reboiler area Areb 26.83 m 2
Top pressure Ptop 4 ·10 5 Pa
chaptersflexibility will be bothassessed forthe debutanizer col-umnasshowninthepreviousparagraphandforadistillation col-umntobedesigned;theresultswillbethencompared.
Moreover,theanalysiswillbeconductedseparatelyforthe de-terministic andstochastic indexes in orderto highlight analogies anddifferences.
3.1. Deterministicindexes:SwaneyGrossmannFSGandresilience
indexRI
First of all, in order to evaluate the Swaney and Grossmann flexibility indexFSG, we need to estimate the variation ranges of
Table 4
Uncertain parameters θk .
Parameter θ Symbol Value θN Expected deviation θk± Unit
Butane flowrate F4 6.863 ± 10% ± 0.686 mol/s
Pentane flowrate F5 2.743 ± 10% ± 0.274 mol/s
Condenser heat transfer coefficient Ucond 473.77 ± 10% ± 47.377 W/m 2 / K
Inlet cooling water temperature Tw 20 ± 10% ± 2 ◦C
Reboiler heat transfer coefficient Ureb 552.90 ± 10% ± 55.29 W/m 2 / K
Max vapor velocity Gf 0.38 ± 10% ± 0.038 m/s
Table 5
Flexibility analysis results: F SG .
Deviation [%] Condenser area [ m 2 ] Reboiler area [ m 2 ] Minimum diameter max [ m ] Maximum diameter min [ m ]
0.00 32.91 22.26 0.603 0.782 5.00 38.08 24.59 0.634 0.744 6.66 40.00 25.43 0.644 0.732 9.35 43.35 26.83 0.662 0.712 15.00 51.53 30.09 0.701 0.673 21.00 62.30 34.05 0.746 0.632
Fig. 6. Heat transfer areas ( F SG ).
theuncertain variablesasshown in Table 4,calledfromhere on out“expecteddeviations”.
Since the system is rather simple (i.e. convex feasibility do-main),forthemomentthere’snoneedofoutliningthewhole fea-siblespace as discussed in the first section; the vertex analysis, by increasing the parameters variation percentages until the ex-pectedvaluesandkeepingconstanttheratios betweenthem,can betheneasilyandeffectivelyperformed.Sincethereare7 chang-ingparameters,thehyperrectangleshas7dimensionsthatmeans 27=128verticesshouldbetheoreticallycalculatedforeach
simu-lationbut,thankstothepossibilitytodecouplesomeindependent parameters,thecomputationaleffortcanbesubstantiallyreduced. The completeflexibility analysisresultsandthecorresponding plotsareshownrespectivelyin Table5and Figs.6and 7,the debu-tanizerrelatedvaluesareindicated.
The resulting Swaney andGrossmann flexibility indexfor this casestudyisthengivenby:
FSG=
allowedde
v
iation expecteddev
iation=5%
10%=0.5 (3.1)
Thebottleneckof flexibilityis givenbythe columnminimum di-ameterequalto0.634m, i.e.by thefloodingconditions.However, theanalysishasbeen carriedout forallthe other design param-etersstandalone aswell. The condenser resultsto be the second mostconstrainingvariable,whilethe reboilerthethirdone;fora value between 12 and 15%of the allowed deviation we can no-tice that the maximum diameter becomes lower than the mini-mumonecausingthecolumndesigntobeimpossiblewithasingle diametercolumn. Thislastphenomenonoccursbecauseflexibility indexesrefer both topositive and negativeperturbations causing therangeofdiametervaluesabletoensureoperableconditionsto becomesmallerforhigherflexibilityrequirements.
Fig. 7. Column max & min diameters ( F SG ).
Theconditionsthatcausethehyperrectangletoexceedthe fea-sibility domain boundaries, i.e. which vertex is tangent one, for each designvariableare reportedin Table 6;“+” and“-” indicate respectively a positive ornegativedeviation ofthe uncertain pa-rameterwhile“/” indicatesthattheparameterdoesnotaffectthe constrainingdesignvariable.
For the“Condenser area”, “Reboiler area” and “Minimum col-umndiameter” critical conditionsareachievedbecauseofoverfed columnandunderperforming equipmentwhilefor themaximum diameter,i.e.weepingconditions,criticalitiesarepresentincaseof underfeedingasexpectedaccordingtothephysicsoftheproblem. Besidethedesignandsizing,aneconomicassessmenthasbeen performedaswell.Thepurposeistocoupleflexibilityand invest-ment costs inorder tomake thebest decisionduringthe design phase and avoid the plant underperform; operational costs have not beentakeninto account sincethey’re univoquely determined dueto thefact that we’re notchanging the numberoftrays.For moredetailaboutequipmentcostscorrelationscf. AppendixB
The capital costs trends (normalized to the lowest value) as function of FSG flexibility are plotted for each equipment in
Figure 8. All of them increase as flexibility increases (according withtheir size). The most expensiveequipment is the Kettle re-boiler that hasa much more accurate technology than theother heatexchangersandthatworksunderpressure,whilethecolumn isrelativelycheapbecauseofitssmalldiameter.
In Fig.9threeseriesofpercentagedataasfunctionofflexibility havebeenplotted;theyrefernamelyto:
• dCC
0:theadditionalinvestmentreferredtoacolumndesignedin nominaloperatingconditions,i.e.0%flexibility(Bluetrend);
• CdC
e f f:thecostdifferenceswithrespecttothecasestudycolumn
Table 6 Critical vertices.
Parameter vs design Condenser area Reboiler area Minimum diameter Maximum diameter
Butane flowrate + + + – Pentane flowrate + + + – UCond – / / / Tw + / / / UReb / – / / Gf / / – / Gw / / / +
Fig. 8. Equipments bare module cost ( F SG ).
Fig. 9. Capital costs comparison ( F SG ).
• CdC
real: theeffectivecost differencesifthe casestudycolumnis alreadyavailable(Yellowtrend).
We can notice first that the trends are more or less linear; moreover,wecanalsopointoutthatpartoftheinvestment(about 3%)could have beensavedif thereboiler wasproperlydesigned, froma flexibility point ofview,since its overdesignis practically useless considering thebottleneck of 5.00% given by the column diameter.
DifferentlyfromtheFSG index,theresilienceindexdefinesthe
largesttotaldisturbance loadasystemis abletowithstand inde-pendentlyofthedirectionofthedisturbance.
It wasoriginally definedforheat exchangers networks, there-foresome modifications areneeded inorderto adaptit toevery type of system. First of all the Resilience Index hasa dimension thatusuallyisan “heat” relatedvalue(kW,Ketc.);inthiscase,in ordertomakethecomparisonwiththeotherindexespossibleand becauseof the severalperturbations we have to copewith, each one with its different dimension, the percentagedeviation value willbe used.Then,sincewe’re dealingwitha simplesystem(i.e. quasi-convexdomain), we’resurewe’re goingtosolve asocalled “Class1problem”, i.e.the standardvertexanalysisprocedurecan beperformedsuccessfully.
ThecompleteResilienceanalysisresultsare shownin Tables 7 and 8and Figs.10and 11.
The resulting Resilience Index(RI) for this casestudy isthen givenby: RI=min i
|
lmax i|
=9.64 (3.2)First ofall we can noticethat there are noproblems even ifthe intervalsareonlyhalf-boundedbecausewe’relookingforthe min-imum of the maximum withstood perturbations. The ∞ values don’t mean that we really calculated results forinfinite percent-agevaluesofparameterdeviation,butitmeansthatthewithstood percentageperturbationislarge enough notto affectthe flexibil-ityanalysisorenoughtofulfillallthephysicallypossible(notonly expected)deviationrangeinthatdirection.
Then wecan focusour attentiononthelimitingdesign factor thatis,asforFSGindex,thecolumndiameterwhosecorresponding
parameteristhefloodingvelocity.TheResilienceIndexcalculated thiswayhasanhighervaluethantheFSGsinceweperturbateonly
oneparameterattimeleavingunchangedtheothers.Viceversa,in orderto attaina givenflexibility value, asmaller oversizing than
FSGcaseisneeded.
Even the crossover of the minimum and maximum diameter values,i.e. the completely infeasible conditions witha single di-ametercolumn,wasmoreconservativeintheFSGanalysiswhereit
wasabout13%(cf. Fig.7),whileintheRIthisconditionisattained fora24%(cf. Fig.11)flexibilitymoreorless.
Moreover,itisworthnoticingthat,besidethecolumndiameter, thesecondmostconstrainingvariableisthereboilerwhileforthe
FSGanalysiswasthecondenser.Thisisthemostrepresentative
dif-ference betweenthetwo indexesbecausethe variables actingon thecondenserareUcond andTw inbothcasesbut,whileintheRI
analysistheyareperturbatedoneatatime,inSwaneyand Gross-mannanalysisthey changeall atoncecausingtheequipment af-fectedbyan highernumberofuncertainparameters (inthiscase thecondenser)tobemorecritical.
As well as the Swaney and Grossmann index costs analysis, thecapitalcosts trends(normalized tothelowest value)as func-tion of RI flexibility are plotted for each equipment in Fig. 12. All ofthem increase asflexibility increases(according withtheir size).ThemostexpensiveequipmentistheKettlereboilerthathas
Table 7
Allowed disturbances loads.
Parameter Maximum deviation [%] l max+
i Minimum deviation [%] l max−
i Limiting constraint
Butane flowrate 11.80 ∞ Dmin
Pentane flowrate 81.00 ∞ Dmin
Inlet cooling water temperature 18.28 ∞ Acond
Condenser heat transfer coefficient ∞ 17.74 Acond
Reboiler heat transfer coefficient ∞ 17.02 Areb
Max vapor velocity ∞ 9.64 Dmin
Min vapor velocity 52.18 ∞ Dmax
Table 8
Flexibility analysis results: RI .
Deviation [%] Condenser area [ m 2 ] Reboiler area [ m 2 ] Minimum diameter max [ m ] Maximum diameter min [ m ]
0.00 32.91 22.26 0.603 0.782
9.64 36.42 24.64 0.634 0.745
17.02 39.65 26.82 0.662 0,715
17.74 40.00 27.06 0.665 0.712
24.00 43.30 29.29 0.691 0.686
Fig. 10. Heat transfer areas ( RI ).
Fig. 11. Column max & min diameters ( RI ).
Fig. 12. Equipments bare module cost ( RI ).
a much more accurate technology than the other heat exchang-ersandthat worksunderpressure,whilethecolumnisrelatively cheapbecauseofitssmalldiameter.
In Fig.13thethreeseriesofpercentagedataasfunctionof Re-silienceIndexanalogoustothosein Fig.9arereported.
Eveninthiscasethetrendsaremorelinear,thedifferences be-tween FSG and RIindexes economical analysis reflects the
differ-enceshighlightedintheequipmentdesignanalysisaswellasthe analogies, i.e.foragivenflexibility valuethecapitalcost islower ifweconsidertheResilienceIndexflexibility.Obviouslyitdoesn’t meanthat wecansavemoneyjustbychangingtheindexweuse, itonlymeansthatbyselectingadifferentindexwearemeasuring differentperformances;soforeach processthemoresuitable the indexisthemoreaccurate theeconomicalanalysiswillbe,where withtheword“suitable” referstotheperformancesdemandedto copewiththepossibledisturbances.
The Swaney and Grossmann flexibility measures the ability of the system to withstand an overall parameters deviation of
FSG·
θ
k±%, the Resilience Index measures the case of asin-gle variable, it doesn’t mind which one, deviation of RI%, there-fore, in order to find the mostsuitable index, we need to eval-uate which of the two kinds of deviation our system is more likely to undergo to. This idea of “disturbance likelihood” leads
Fig. 13. Capital costs comparison ( RI ).
usthentothestochastic characterizationoftheflexibilityindexes discussed in thefollowing section forthe same debutanizer case study.
3.2. Stochasticindexes:PistikopoulosandMazzuchiSFandLaiand HuiFV
Whenwemovefromadeterministicpointofviewtoa stochas-ticonewehavetotransformtheideaofperturbationexpectedor not,definedby awellboundeddeviationrange,intoacontinuous functiondescribinghowmuchexpectedthedeviationis.Todothis weneedtouseaprobabilitydistributionfunctionrelatinga prob-abilityvaluetoeachcondition“x”,i.e.theindependentvariable(s) (cf. AppendixCformoredetails).
The probability density function are usually parametric func-tions, theprobability ofthe independentvariablefallingwithin a particularrangeofvaluesisgivenbytheintegral ofthisvariable’ sPDFoverthatrange.Theprobabilitydensityfunctionis nonneg-ativeeverywhereanditsintegral overtheentirespaceisequalto one.
Beforegoinganyfurtherwiththeapplicationoftheseprinciples toourdebutanizercasestudy,afewobservationsaboutthechosen PDFanditspropertieswillberemarkedherebelow.
First ofall, inorder tounivocally define thePDFs we needto settheparameters;theselectedvaluesare:
•
μ
=operatingconditionsforeachvariable;•
σ
orb=20%ofμ
foreachvariable.For thiscase study a 20% variance has been selected since it is aboutthe maximum allowed individual deviationfor the vari-ables takenintoaccount inthestochastic flexibilityanalysis.This choice makes the analysissensibleenough to appreciatethe fea-turesofthecouplingofflexibilityandeconomics,thatisthemain goalofthe paper.Averysmall
σ
value wouldresultin auseless flexibilityanalysissincealmostallpossibleperturbationswouldbe withstood;ontheotherhandahigherσ
valuehaspoorreliability sincevariablesuncertaintyrangewouldbetoowideanda signifi-cantstochasticflexibilityvaluewouldbeneverattained.Therefore 20%resultsagoodcompromiseallowingananalysismoresensible andunbiasedbyexcessiveoptimism,keepingusasconservativeas needed.We havethento choosewhichtype ofPDFswouldbetter de-scribe the systemunder analysis.Probability reflects the state of theinformationtherefore,sinceweactuallyhavenodataaboutthe
probability functionsof ourparametersdeviations, themost gen-eralPDF possibleshouldbeusedinordertohavethemore unbi-asedpossibleresults.Theconditionof“generalvalidity” issatisfied bytheGaussianornormalprobabilitydistribution.Itis symmetri-calwithrespecttoits meanandthe99.73%ofcumulative proba-bilityfallsintherange[−3
σ
,+3σ
].Forthesakeofcompleteness,inordertoprovethattheresults of the stochastic flexibility analysis are not qualitatively PDF de-pendent,itwillbeconductedwithadifferentprobability distribu-tionfunctionaswell,i.e.theLaplaceone.Thisdistributionsatisfies, inasense,thesamerequirementsneededbythesystem descrip-tion,thatis:
• The maximum probability is attained at the operating condi-tions;
• the probabilityvalue is notdependent onthedeviation direc-tion.
ForfurtherdetailsaboutNormalandLaplaceprobability distri-butionfunctionscf. AppendixC.
Inordertohaveavisual approachwiththe stochastic flexibil-ityindex meaning, a 2D analysis forthe condenser perturbation hasbeenperformedfirst(Figs.14 and 15).Then, oncedealtwith it, the analysisis shifted to higher dimensions; it is nonetheless worthremarkingthattheanalysismethodologyisindependenton thedimensionoftheproblem.Aswecannotice,thetwoselected parameters,i.e.heattransfercoefficientandcoolingwater temper-ature,actonthesamedesignvariable,i.e.heatexchangertransfer area.Thereforeasingleconstraintrepresentingtheheatbalanceis present.
However,sincewedon’twanttoperformtheflexibilityanalysis ofaheatexchanger,ononehandweneedtoincreasethe dimen-sionof our problemon the other handwe don’t wantthe com-putationalefforttobe toohigh. Tomatchthesetwopurposeswe could,forinstance,add themostconstrainingparameter (accord-ing to the previous flexibility analysis), i.e.the flooding velocity. Thiswaywehavea3D domainwithavariable(Gf) relatedtothe
columndesign (Dmin)andtheother two(Tw andUcond)actingon
thesamedesignvariable(Acond)(Fig.16);theindependenceofthe
thirdparameterontheothertwocanbeimmediatelynoticedsince theyellowplane,i.e.floodingconstraint,isparalleltotheTwxUcond
plane.
An additional difference betweenthis index andthe previous oneswe’vegottodealwithisthat,evenifagivendesigndefines univocallyastochasticflexibilityvalue,agivenstochasticflexibility value does not define univocally a system design. The two con-straints can be shifted in several configuration keeping constant thevalue ofthe integral functionindeed.Therefore we’ll referas the“x” stochasticflexibilityvaluedesigntotheoptimal configura-tionthatattainsthatvalue, whereoptimalsimplymeanscheaper. Thisneedofeconomicaloptimizationdirectlylinksflexibilitywith design and economic aspects whose trends will be anticipated withrespecttothesizingones.
Theresultsofthestochasticflexibilityanalysisforeach1%cost increase arereportedin Figs.17 and 18,theoptimaldesign vari-ablesin Figs.19and 20.
First ofall it’s worth highlighting that, differentlyfrom previ-ousflexibility indexes,thestochastic one hasanon-zerovalue at operatingconditionsdesign.
Moreover the optimal design accordingto flexibility could be differentfromthe operatingconditions design orthe economical optimaldesign.
An additional difference between SF andFSG or RI is that its
trend is highlynon-linear if expressed as function of the equip-ments’size,thatmeansthatforbigoversizedequipmentthe flexi-bilityincreaseduetoafurtheroversizingisonlyslightly apprecia-ble.The relativeoptimal oversizing trendbetweenthe condenser
Fig. 14. Condenser SF : Normal PDF.
Fig. 15. Condenser SF : Laplace PDF.
Fig. 16. 3D SF analysis domain.
andthecolumnkeepsbeingnonethelessalmostlinearand further-moreitfollowsthesamelineindependentlyonthestartingdesign conditionasexpected.
Finallyifwecomparethetwodifferentdistributionsresultswe canimmediatelynoticethattheyreflectthenatureofthe distribu-tionsthemselves.Laplacedistributionconvergesmoreslowlythan theNormal one, therefore the relatedSF approachesthe value 1 onlyforabiggeroversizing. Nonetheless theirtrends are qualita-tivelyassimilarasthePDFare.
Fig. 17. SF results normal PDF.
Thesamesizingrelatedremarksarevalidifwetalkaboutcosts sincethey’redirectlyrelated.
Fortwo differentequipment design the startingpointsof the
dC
C
v
s.SF linesaredifferentbut,afterawhile,theendingbranchesofthetwocurvesoverlapeachotherapproachingtheasymptote. Additional costs are higher for the Laplace distribution case thanNormaldistribution,asexpected,becausethedeviation like-hoodisslightlyhigherevenfarfromtheoperatingconditions.
Fig. 18. SF results Laplace PDF.
Fig. 19. SF optimal sizing NDF.
Fig. 20. SF optimal sizing LDF.
Fig. 21. dC
Cv s. SF derivatives ratio.
Differently from FSG and RI, costs increase as function of
stochastic flexibility index shows a non-linear trend, this means that once a certain point ofthe curve hasbeen passedthe ratio betweentheincreaseinflexibilityandcostsstartsdecliningfast.
ThemainpurposeofProcessSystemEngineeringistoenhance thedecisionmakingcapabilityofthechemical engineer,therefore we’dliketoidentifythe“certainpoint” afterwhomit’snotworth keepingspending money in overdesign. The range of convenient oversizingisvisibletotheunaidedeye:thefirstpartofthecurve givesa highflexibility increase witha smalladditional costs but theunwithstooddeviationprobabilityisstillrelevant;ontheother handthe last part,even withstanding almost the whole possible deviations,needs a consistent oversizing (i.e. additional costs) to be achieved.Finallywe canthen concludethat the middlerange of the curve is a good compromise between high flexibility and affordableadditionalcost.
Thethoroughprocedureweproposetoassesstheoptimalrange is based on the curve properties: tangent lines have almost the sameslope(i.e.derivative)atthebeginningandattheendofthe curve whileit considerably changeswithin the interval we’re in-terestedin.
Theprocedure then consistsin plottingtheratio betweenthe derivativecalculatedateachpointandthederivativecalculatedat theprevious oneasillustrated in Fig.21 forthe caseofNDFand operatingconditions.
Thisway,giventhe dC
C
v
s.SFplotonly,wecanobtainanewplotwhosetrend showsamaximum corresponding tothe value SF= 0.9409,i.e.10%ofadditionalcost.
However even the other values nearthere can be considered goodconditionsforaflexibilitybaseddesign,moreimportantthan theoptimalvalueitselfistoavoidtheconditionscorrespondingto very first and very last part of the plot. The chronologically last flexibility indexproposed inscientific literature isthe volumetric flexibilityindexFV.Itisdefinedasthefeasiblefractionofthe
un-certain space, that is a linefor 1D case, a surfacefor2D case, a volumefor3Dcaseandahypervolumeforhigherdimensions.
The reason why the volumetric flexibility index is included amongthestochasticindexesisthatitcanbealsothoughtasaSF
indexparticular case whosedeviation probability function is de-scribedbyastepfunction(Fig.22)definedas:
1 N i=1(
θ
iU−θ
iL)
if|
xi|
<θ
i± 0 if|
xi|
>θ
i± (3.3)Fig. 22. F V probability distribution function.
Fig. 23. 2D F V analysis.
Thus,inordertoperformthiskindofanalysis,twoparameters havetobeset:thenominalpointtotheoperatingconditionsand theexpecteddeviationvalueto20%.
Forthisindexaswella2Danalysisforthecondenser perturba-tionhasbeenperformedfirst(Fig.23)andthen,oncedealtwithit, theanalysisisshiftedto higherdimensions. ThesameSFanalysis remarksapply,i.e.theanalysismethodologyisindependentonthe dimensionoftheproblem,thetwoselectedparametersactonthe samedesignvariable,i.e.heatexchangertransferarea,thereforea singleconstraintrepresentingtheheatbalanceispresent.
The resultsofthis2DcasestudyareFV =0.4714foroperating
conditionsandFV =0.8584forrealconditions.
For theFV indextheanalysishas beenshifted inthesame SF
3Ddomain(Fig.24).Thefloodingvelocityhasthenbeenincluded intheanalysis;sinceit isthemostconstrainingparameter,it re-ducessignificantlythefeasibilityofthesystem.Thetripleintegral requiredbytheSFindexactuallybecomesavolumeintegralsince theprobabilitydistributionisconstantoneachintervalwhereitis defined;duetothecomplexityofPDFtobeintegratedin stochas-ticflexibility analysis,thecomputationaleffortforthe FV analysis
provemuchlower(secondsvs.minutes).
Even inthiscase agiven volumetricflexibility value doesnot define univocally a system design, therefore the “x” volumetric flexibilityvaluedesignreferstotheoptimal,i.e.cheapest, configu-rationthatattainsthatvalue.
Fig. 24. 3D F V analysis domain - operating conditions.
The results ofthe volumetric flexibility analysisfor each 0.5% costincreaseareshownin Figs.25and 26.
On one hand we have the analogies with the SF that are non-zero values at operating conditions, optimal sizing inde-pendent on the starting conditions and asymptotes at FV =1.
On the other hand we’ve got to notice that the trends of the
FV analysis results are rather peculiar and very distinctive of
this kind of index. In Fig. 26 we can mainly distinguish three zones:
1. Lineartrend:Intheveryfirstoverdesignpartboththecolumn diameterandcondenserareaaffecttheflexibilityofthesystem, thereforetheoptimaldesignstrategyistosplittheinvestment accordingtotheproportionexpressedbytheslopeoftheline. 2. Horizontal (or vertical)trend: After a whilethe flooding
con-straintisalmostoutsidetheuncertainparallelepiped.Theonly thingtodoisthentoshiftthecondenserconstraintuntilit ex-itstheparallelepipedaswell. Intheendalittleadjustmentof both the design variables is observed andthe value FV =1 is
finallyattained.
3. Overpayment zone: After the FV =1 value has been achieved
the whole uncertain domain is contained within the feasible boundaries therefore any additional oversizing is, considering flexibility,practically useless. The Acond=50.90m2 and Dcol=
0.686mconditioncorrespondsdefinitelythemaximum achiev-ableflexibility.
Fig. 25. F V analysis economic results.
Fig. 26. F V analysis optimal sizing.
In the end it is worth remarking that the shape of the over-sizing curve is strictly related to the system features and to the expecteddeviations,theflexibilityanalysisprocedurehas nonethe-lessgeneralvalidity.
The cost related remarks are the direct consequence of the combinationofsizesrelatedones.Theincreaseininvestmentcosts asfunctionofvolumetricflexibility showsan almost lineartrend. Since theequivalentprobabilitydistribution hasaconstant value, the FV indextendency resultsmore similar to theFSG andRI
in-dexes one, even ifthe domain considerations, the waywe com-pute its value andthe propertiesofthe oversizing curves practi-callymakeitanalogoustotheSFindex.
In theend we can concludethat a 5.5% additionalcost (w.r.t. operatingconditions)isdefinitelythemaximuminvestmentworth tobedoneaccordingtoaflexibilityimprovementpurpose.
4. Conclusion
After these four flexibility analysis an indexes comparison is necessary to summarizeandcomment therelationships between alltheresults.
Firstofallwecanmainlydistinguishtwotypologiesofindexes accordingto their approachtowards the feasibledomain: the in-dexesassessing the minimumof themaximum performance and theonesgivingaglobalassessmentwithin theuncertainor feasi-bledomain.
TheFSG andRIindexesaregenerallymoreconservative,they’re
basedonn-Dhyperpoligonsthatcan beinscribedwithinthe fea-sibledomain,wherenisthedimensionoftheuncertainzone(i.e. the numberof possibly perturbatedparameters). The first one is more suitable ifthe expectedperturbation involves several vari-ables at the sametime, the second one for biggervariations re-ferred to single variables. However, they’re both very conserva-tive(speciallyFSG)sincethedescribedgeometricalobjectscanonly
scaleupordownbuttheirstructureisnotflexibleatall,itdoesn’t coastthedomain,itjuststopswhetherone pointistangent.This waytwocompletely differentsystemscould havethe same flexi-bilityindex value justbecause they allow the same perturbation intensityfor themost constrainingparameter only; onthe other handgivenacertainstartingdesign,aflexibilityindexdefines uni-vocallythesystemconfiguration.
Ontheotherhandwehavemoreoptimisticindexeswith non-null value at operating conditions that require integration to be calculated:SF andFV.Evenin thiscaseone ofthem refers toan
expecteddeviationwhilethe other one needs an apriori knowl-edgeorestimationofperturbationlikelihood.FV resultsare
simi-lartothelinearonesuntiltheindexachievesitsmaximumvalue of1afterwhomevery kindofoversizingisuseless;ontheother hand the SF index is very sensitive and smart, it takes into ac-count all the possible deviations proportionally to their likeli-hood,showinganhyperbolictrendinoversizing/costvs.flexibility plots that reflects well the real capability of the system to bet-ter withstand small and likely deviations than big and unlikely ones. Several configurations may have the same SF or FV index
valuesincetheyassessaglobalsystemproperty,notthemost con-strainingone;thereforeduringtheflexibilityassessmentwecould need to solve an optimization problemaccording to the analysis purpose.
In the endit’s worth remarking that the value of FSG and FV
strictly dependsonthe expecteddeviations, neverthelessFSG can
be generalized and compared to the whole feasible domain as shownwhileFV cannot.
Then, froma designpoint ofview we can definitely conclude thatthemostsuitable flexibilityindexforthe particularanalyzed systemisafunctionoftheexpecteddeviationnature.
Anadvice ofgeneralvalidityistoperform theflexibility anal-ysisusingmorethanone index,combiningthiswaytheir advan-tages;forinstancethefirsttwoindexesarequiteeasytocompute whiletheonesrequiringanintegralcalculationhaveahigher com-putationaleffortdemand.Thuswecanperformtheflexibility anal-ysiswithFSG orRIfirstinorderto identifythemostconstraining
variablesinordertobeabletoreducethedimensionofthe prob-lemandperform the SForFV analysison asmaller domain.
Ob-viouslythisisagoodprocedureifsomeparametersperturbations affectvariables whoseconstraints are very loose; ifthe order of magnitudeofalltheFSGiorRIiisalmostthesame,thispreliminary
analysisisoflittlehelpsincewecannotbesurethatthemost con-strainingvariableforone indexwillbe themostconstrainingfor theother indexesaswell unlessmuchlessrelevantdeviationsare allowedwithrespecttotheothers.
In order to compare the four indexes more immediately, the Table9resumingalltheirmainfeatureshasbeenoutlined.
On the other hand,from an economic perspective, the intro-ductionoftheCapitalcostsvs.Flexibilityrelationshipandits rela-tiveplotsletthedecisionmaker,i.e.theengineering,take amore informed decision whatever the adopted flexibility index. More-over,itclearlyshowshowanaprioriflexibilityanalysisduringthe
Table 9
Steady state flexibility indexes comparison.
Pros/Cons FSG RI SF FV
Need for data Expected deviation − + + −
Feasibility region outline −/+ −/+ − −
Probability distribution −
Bounded domain −/+ −/+ + +
Computational effort Vertex analysis + + − −
Computational effort + + + 4 + + +
Results accuracy Conservative − −/+ + +
Accuracy − −/+ + −/+
design phase could have allowed to save part ofthe investment keepingunchangedtheflexibilityperformanceofthesystem.
AppendixA. Listofacronymsandsymbols
Symbol Definition Unit
A Characteristic dimension m n
Acond Condenser heat transfer area m 2
CBM Equipment bare module cost $
C0
p Purchase equipment cost in base conditions $
CDF Cumulative distribution function Function
d Design variables /
Dcol Column diameter m
Dmin Minimum column diameter m
DF Dynamic flexibility index 1
fj Inequality constraint Function
F4 Butane partial flowrate mol/s
F5 Pentane partial flowrate mol/s
FBM Bare module factor 1
FM Material factor 1
FP Pressure factor 1
Fq Column trays factor 1
FSG Swaney and Grossmann flexibility index 1
FV Lai & Hui flexibility index 1
Gf Flooding velocity m/s
Gw Weeping velocity m/s
h Equality constraint Function
lmax±
i , l ±i (Maximum) i-th direction disturbance load 1
LDF Laplace distribution function Function
M& S Marshall & Swift cost index 1
N, N r , N s Overall, rectifying and stripping stages 1
NDF Normal distribution function Function
P Pressure Pa
Pr Probability 1
PDF Probability distribution function Function
RI Resilience index 1
Se Polygonal feasible space approximation /
Sf Feasible space /
SF Stochastic flexibility index 1
T FSG n-D hyperrectangle /
Tw Inlet cooling water temperature ◦, C
Ucond Condenser heat transfer coefficient W/m 2 / K
Ureb Reboiler heat transfer coefficient W/m 2 / K
V0 FV uncertain volume /
Ve Polygonal feasible volume approximation /
Vf Feasible volume /
z Control variables /
Z Control variables space /
zU , z L Control variables upper and lower limits /
Greek letters Definition Unit
δ hyperrectangle vs. expected deviation ratio 1
δk k-th rectangle ratio 1
θk Uncertain parameters values /
θN Nominal conditions /
θk± Expected deviation 1
θiU , θiL FV upper and lower expected deviation 1
μ PDF mean value 1
σ PDF variance 1
χ Most restrictive dynamic constraint Function
Feasible space /
AppendixB.Capitalcostsestimations
Inordertoevaluatetheinvestmentcostrequiredforthewhole systemormakeanykindofeconomicconsiderationand compari-son,weneedtoestimatethecostsofeverysingleequipment.
ForthispurposetheGuthrie–Ulrich–Navarretecorrelations de-scribedinthenext paragraphs willbeused(Guthrie,1969;1974; NavarreteandCole,2001;Ulrich,1984).
B.1. Purchaseequipmentcostinbaseconditions
Thepurchaseequipmentcostinbaseconditionsisobtainedby meanofthefollowingequation:
log10
(
CP0[$])
=K1+K2· log10(
A)
+K3· [log10(
A)
]2 (B.1)whereAis thecharacteristicdimension adthe Ki coefficientsare
relativetotheequipmenttypology(cf. Table10).
Theprovidedcoefficientsreferstotheyear2001andtoaM&S indexequalto1110.Inordertoupdatethecostsvaluetotheyear 2016 we’ll refer to a M&Sindexequal to 1245.2by mean ofthe correlation: C0 P,2= M&S2 M&S1 · C0 P,1 (B.2)
B.2. Baremodulecost
Theequipmentbaremodulecostcanbecalculatedaccordingto thefollowingcorrelation:
CBM=CP0· FBM (B.3)
wherethebaremodulefactorisgivenby:
FBM=B1+B2· FM· FP (B.4)
The FM andFP factorsrefers to theactual constructionsmaterials
andoperatingpressurewhiletheBicoefficientsreferstothe
equip-menttypology(cf. Table11). TheFP,Kettlevalueisgivenby:
log10
(
FP)
=0.03881− 0.11272· log10(
P)
+0.08183· [log10(
P)
]2(B.5)
wherePistherelativepressurein105 Pascal.
Forcolumntraysbaremodulecost aslightlydifferent correla-tionshouldbeused:
CBM=N· CP0· FBM · Fq (B.6)
whereNistherealtraysnumber,FBM=1eFqisgivenbythe
cor-relation:
log10
(
Fq)
=0.4771+0.08561· log10(
N)
− 0.3473· [log10(
N)
]2i f N<20 (B.7)
Fq=1 i f N≥ 20 (B.8)
AppendixC. Probabilitydistributions
C.1. Normalprobabilitydistributionfunction
Asalreadymentioned,theconditionof“generalvalidity” is rep-resented by the gaussian or normal probability distribution. It is symmetricalwithrespect toits meanandthe 99.73%of cumula-tiveprobabilityfallsintherange[−3
σ
,+3σ
].ThesinglevariableNormalPDF(Fig.27)statesas:
P
(
x)
= 1σ
√2π
e−(x−μ)2
2σ2 (C.1)
Table 10
Equipment cost in base conditions parameters.
Equipment Typology K1 K2 K3 A
Heat exchanger Fixed tubes 4.3247 −0.3030 0.1634 Heat tranfer area [ m 2 ]
Kettle 4.4646 −0.5277 0.3955 Heat transfer area [ m 2 ]
Columns (vessel) Packed/tray 3.4974 0.4485 0.1074 Volume [ m 3 ]
Trays Sieved 2.9949 0.4465 0.3961 Cross sectional area [ m 2 ]
Table 11
Bare module parameters.
Equipment Typology B1 B2 FM FP
Heat exchanger Fixed tubes 1.63 1.66 1 1
Kettle 1.63 1.66 1 FP,Kettle
Columns/vessel / 2.25 1.82 1 1
Pumps Centrifugal 1.89 1.35 1.5 1
Fig. 27. 1 D NDF and its CDF.
Fig. 28. 2 D NDF.
The bivariate Normal PDF (Fig. 28), for a correlation between thevariables
ρ
=0,statesas:P
(
x)
= 1 2πσ
1σ
2e−y2 (C.2)
whereyisgivenby:
y=
(
x1−μ
1)
2σ
2 1 +(
x2−μ
2)
2σ
2 2 (C.3)Finallythe mostgeneral n-variables normal PDF with can be definedas: P
(
x)
=(
21π
)
n/2·|
|
−1 2· e− 1 2·(x−μ) −1(x−μ) (C.4)where
is the variance-covariance matrix, −1 its inverse and |
|itsdeterminant.
Moreoverwecanstandardize,i.e.reconducttoa0meanvalue andvariance equal to 1(variance-covariance matrix equal to the identitymatrix),thenormaldistributionby meanofthe indepen-dentvariablesubstitution:
z=x−
μ
σ
(C.5) obtaining: P(
z)
=(
21π
)
n/2·|
I|
−1 2· e− 1 2·z·I·z (C.6)fora generaln variablesstandard normal probability distribution (Severini,2011).
Thistransformation besides making thecalculations easier al-lowstocomparevariableswithdifferentdimensions,e.g. tempera-turevs.flowratevs.velocityetc.
The boundaries ofthe feasibility domain,if analytically avail-able,havethentoberewrittenasfunctionsofthenewvariablez byinvertingthe Eq.(C.5).
C.2. Laplacedistributionfunction
Theseconddistributionfunctionusedforthestochastic flexibil-ityanalysisisthesocalledLaplacePDF.
This distribution satisfies, in a sense, the same requirements neededbythesystemdescription,thatis:
• The maximum probability is attained at the operating condi-tions;
• the probabilityvalue is notdependent onthedeviation direc-tion.
The analytical expression of the single variable Laplace PDF (Figure29)statesas:
P
(
x)
= 1 2· be−|x−μ|
b (C.7)
where
μ
isthemeanandbisthediversitylocalparameter. Evenin this casewe can standardize the distribution, i.e. re-conducttoa 0meanvalueanddiversityparameterequalto1by meanoftheindependentvariablesubstitution:z=x−
σ
μ
(C.8)obtaining:
P
(
z)
=1 2e−|z| (C.9)
Moreover,keepinginmindthat:
Fig. 29. 1 D Laplace DF and its CDF.
Fig. 30. 2 D Laplace DF.
the trivariate Laplace PDF (for the bivariate cf. Fig. 30), for a correlationbetweenthevariables
ρ
=0,statesas:P
(
x)
= 1 8π
e−√x21+x22+x2
3 (C.11)
The Laplace distribution function converges more slowly than thenormaldistribution,thereforeweexpectalowerflexibility in-creasebyincreasingthesizing,i.e.thecosts.
Supplementarymaterial
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.compchemeng.2019. 02.004.
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