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On the Riemannian structure of the residue curves maps
Nataliya Shcherbakova, Vincent Gerbaud, Ivonne Rodriguez-Donis
To cite this version:
Nataliya Shcherbakova, Vincent Gerbaud, Ivonne Rodriguez-Donis. On the Riemannian structure of
the residue curves maps. Chemical Engineering Research and Design, Elsevier, 2015, vol. 99, pp.
87-96. �10.1016/j.cherd.2015.05.029�. �hal-01200753�
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10.1016/j.cherd.2015.05.029
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To cite this version : Shcherbakova, Nataliya and Gerbaud, Vincent
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On the Riemannian structure of the
residue curves maps
. (2015) Chemical Engineering Research and
Design, vol. 99. pp. 87-96. ISSN 0263-8762
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On
the
Riemannian
structure
of
the
residue
curves
maps
Nataliya
Shcherbakova
a,∗,
Vincent
Gerbaud
b,
Ivonne
Rodriguez-Donis
baLaboratoiredeGénieChimique,UniversitéPaulSabatier–INPENSIACET,4,alléeEmileMonso,CS84234,31432
Toulouse,France
bLaboratoiredeGénieChimique,CNRS,4,alléeEmileMonso,CS84234,31432Toulouse,France
Keywords:
Residuecurves Gradientsystems Riemannianmetric
a
b
s
t
r
a
c
t
Inthispaper,werevisethestructureoftheresiduecurvemaps(RCM)theoryofsimple evaporationfromthepointofviewofDifferentialGeometry.RCMarebroadlyusedforthe qualitativeanalysisofdistillationofmulticomponentmixtureswithinthethermodynamic equilibriummodel.Nevertheless,someoftheirbasicpropertiesarestillamatterof discus-sion.Forinstance,thisconcernstheconnectionbetweenRCMandtheassociatedboiling temperaturesurfaceandthetopologicalcharacterizationofthedistillationboundaries.In thispaperweputinevidencetheRiemannianmetrichiddenbehindthethermodynamic equilibriumconditionwrittenintheformofthevanderWaals–Storonkinequation,andwe showthatthedifferentialequationsofresiduecurveshaveformalgradientstructure.We discussthefirstnon-trivialconsequencesofthisfactfortheRCMtheoryofternarymixtures.
1.
Introduction
Thepreliminarydesignofdistillationprocessesforthe sepa-rationofmulticomponentmixturesreliesupontheanalysis of residue curve maps (RCM). Typically RCM can be used forassessing the distillation columnsequence in continu-ousoperationor thestepsequence inbatchoperation,the achievableproductofeachcolumnorstep,thecomposition andtemperaturetrajectoriesintheproducttanksandinthe column.
Indeed,thetopologicalpropertiesofRCMenableto iden-tifyinthecompositionmanifoldsomefeatureslikeazeotropes and distillationboundaries,whose knowledgeisofutmost importance for the choice of a suitable distillation pro-cessanditsdesign.Otherpropertiesrelevantfordistillation processesare displayed interms oftemperature, unidistri-bution and univolatility manifolds. These properties have beensurveyedinseveral works,inparticularinthe review paperbyKiva et al.(2003), whichalso provides a compre-hensivehistoryofRCM, and inDoherty andMalone (2001)
∗ Correspondingauthor.Tel.:+33534323650.
E-mailaddress:nshcherb@ensiacet.fr(N.Shcherbakova).
and Petlyuck books(2004).Theusefulness ofthese proper-tiesforazeotropicdistillationprocessdesignisdescribedin
WidagdoandSeider(1996)andSkiborowskietal.(2014)and for extractive distillation in Gerbaud and Rodriguez-Donis (2014).
Inthispaper,wefocusonthesimpleisobaricdistillationof
homogeneousn-componentmixturesunderthermodynamic
equilibrium.Residuecurvesdescribetheevolutionofthe liq-uidcompositionwithrespecttosomeparameter,andcan becomputed bysolvingthesystem ofordinary differential equations
dxi
d =vi(x,Tb(x1,...,xn−1)), i=1,...,n−1, (1) wherevi=xi−yi,i=1,...,n−1arecomponentsofthe
equilib-riumvectorfieldv,Tbistheboilingtemperatureofthemixture
ofcompositionx=(x1,...,xn−1),xi,yibeingthemolar
concen-trationsoftheithcomponentintheliquidandvapourphases correspondingly. Despiteoftheir broadutilisation,someof the basic properties ofresidue curves are still a matterof
discussion.Indeed,itturnsoutthattheglobalintrinsic struc-tureofRCMisnotyetwellestablished.
Several authors observedthat many features ofresidue curvesmakethink thattheyare integralcurvesofa gradi-entflowassociatedtotheboilingtemperatureTb.Indeed,the
criticalpoints ofTb are singularpoints ofv,which
generi-callycanbestable/unstablenodesorsaddles.Tbisincreasing
alongresiduecurves,moreover,TbisaLyapunovfunctionfor
dynamicalsystem(1).Theseare typicalpropertiesof gradi-ent systems(Hirschet al., 2004),but they arecontradicted bythefactthatinreal mixturesvisnotorthogonaltothe isothermsurfacesofTb.Forinstance,thenon-orthogonality
was shown by van Dongen and Doherty (1984), who
dis-played the boiling temperature surface isotherms and the steepest descent lines along with the corresponding RCM for four azeotropic mixtures. They also proved that sys-tem(1) cannotbewritten asagradient systemofTb. This
result was later confirmed in Rev (1992). It is natural to ask what is then the true intrinsic structure of equations
(1)?
Thenextquestion,closelyrelatedtothepreviousone, con-cernsthenatureofthedistillationboundariesofRCM.Recall thatinthecaseofternarymixtures,distillationboundaries are remarkableresidue curves connecting nodes and sad-dles,whichdividethedistillationdomainindistinctregions. Bytheirnature,the boundariescannotbecrossedbyother residue curves and theymust start and end atthe singu-larpointsoftheRCM.Usuallythedistillationboundariesare computed numerically as separactrices of system (1) with somelossinprecision duetothe numericalintegration. A differentapproach based onthe variational viewpoint was recentlyproposed by Lucia and Taylor (2006,2007). In the caseofternarymixtures,definingdistillationboundariesas theconcatenationofresiduecurvesgoingfromanunstable nodetoastablenodepassingthroughasaddle,theyshowed thatdistillationboundariesmaximize thelengthamongall other residue curves joining the same points. In the case
n=4theyclaimthatdistillationboundariesareminimal sur-faces(BellowsandLucia,2007).Theinterestingandstillopen questioniswhetherdistillationboundariescanbedetected without numerical integration, for instance, by computing somescalarparameterthatdistinguishesthemamongother residuecurves.
Formany yearsit wasa commonbelief thatdistillation boundariesareprojectionsontheRCMplaneoftheflexures oftheboilingtemperaturesurface,theso-calledridge/valley curves.Butitturnsoutthatthispicturecontradictswiththe experimentaldata, andnumericalcomputations(Rev, 1992; vanDongenand Doherty,1984),so that todaymostofthe authorsagreethattheridge/valleycurvesoftheboiling tem-peraturesurfacearenotdistillationboundariessincetheycan becrossedbysomeresiduecurves.Inthiscontextwewantto stressoutthatthereisnoonecommonlyaccepteddefinition oftheridge/valleycurvesoftheboilingtemperature(seein
Kivaetal.,2003andreferencestherein).So,manyofpublished resultsarebasedonratherwronggeometricalconstructions, sometimesleadingtoparadoxicalresults,likethevalleys pre-sentedinvanDongenandDoherty(1984)thatdonotevenpass throughazeotropes.However,thenotionofaridgeoravalley onasurfacehasaclearmathematicalmeaningin Differen-tialGeometry,andinparticularin3DImageProcessdomain (Bruceetal.,1996;PeikertandSadlo,2008).Itseemsimportant toanalyzetheconsistencyofthisnotionwiththedefinition ofdistillationboundaries.
Inthispaper,wetrytoanswerthefollowingnatural ques-tions:
Q1: What isthe relation betweenisotherm hyper-surfaces and residue curves? More generally, what is the true
intrinsicstructureofequations(1)?
Q2: Howfast the boilingtemperature grows along residue curves?
Q3: Whataretheridge/valleycurvesoftheboiling tempera-turesurfaceandisthereanyrelationbetweenthemand thedistillationboundaries?
The key tools of our analysis are the van der
Waals–Storonkin equations of phase coexistence, which
expressthethermodynamicequilibriumconditiondG=0and generalize theclassical vander Waalsequations forbinary mixtures to the multicomponent case (see in Storonkin, 1967;ZharovandSerafimov,1975;ToikkaandJenkins,2002). In the RCM theory these equations imply the remarkable relation betweenthe boiling temperature gradient and the equilibriumvectorfield
∇Tb= 1
1sD
2
xglv, (2)
whichcanbefound,forinstance,inDohertyandMalone(2001)
andDohertyandPerkins(1978).Here1sissomepositivescalar functiondepending onthe molarentropiesand concentra-tionsofeachcomponentinbothphases,whileglistheGibbs
free energyofthe liquid phase. Already in 1970’s, Filippov remarkedthattheHessianoftheGibbsfreeenergyappearing inthecoupleofvanderWaals–Stronkinequationsforboth phasesdefinesametricinthemathematicalsense(Filippov, 1977).Heusedthisfactforthelocalanalysisofthebehavior oftheresiduecurvesinthevicinityoftheinternalazeotropes. Though inadifferentway,wecometoasimilarresultand introducea metric(different fromthe Filippov’sone) asso-ciatedtotheHessian oftheGibbsfreeenergyoftheliquid phase,whichleadsustoaratherfar-goingconclusionabout theglobalgradientnatureofRCMequations(1).
This paperisorganizedas follows.Afterashortreview ofsomebasicfactsfromRiemanniangeometryinSection2, inSection3,usingthevanderWaals–Storonkinequation,we showthattheRCMofopenevaporationcarriesonanon-trivial RiemannianmetricthatwecalltheGibbsmetric.Recallthata metricinthespacedefinesthewaytocomputescalar prod-ucts,andhencenormsandanglesbetweenvectors,aswell asthelengthofcurvesandthegradientsoffunctions.In gen-eral,thestandardEuclideanmetricused“bydefault”givesjust thelocalapproximationofthetruegeometricalstructureofthe space,like,forinstance,thecityplanthatrepresentsasmall pieceoftheEarthglobe.Thepresenceofthenon-trivialGibbs metricallowsustoprovethatsystem(1)isagradientsystemof
Tb,wherethegradientshouldbecomputedinthe
Riemann-iansense.Thisfactexplainstheaforementionedqualitative properties ofRCM, and in addition, it implies that residue curvesareindeedorthogonaltotheisothermhyper-surfaces inthesenseoftheGibbsmetric.InSection4weanalyzein greaterdetailtheternarymixturescase.Inparticular,we con-siderarigorousmathematicaldefinitionofridge/valleycurves oftheboilingtemperatureandshowthatthereisnoreason forthesecurvestocoincidewithdistillationboundaries.We illustrateourcomputationsforternarymixturescombining
analyticalandnumericalcomputationsusingMathematica9 package.
2.
Riemannian
structures
Inthispaper,weconsiderthestatespaceofaphysical sys-temasadifferentialmanifoldM,whosedimensionisequalto thenumberofdegreesoffreedomofthesystem.Theevolution ofthesystemisusuallymeasuredwithrespecttosome non-decreasingscalarparameter,forinstance,timet.Atagiven momentoftime thestateofthe systemwithn degreesof freedomisapointx∈M,whichcanbedescribedbythesetof
localcoordinatesx=(x1,...,xn),whosederivatives ˙x =( ˙x1,..., ˙xn)
form a velocity vector ˙x∈TxM in the tangent space to M
atx.MiscalledaRiemannianmanifoldifitstangentbundle
TM=
S
x∈MTxM isendowed witha scalarproduct. In otherwords,thereisapositivedefinitequadraticformcalledmetric, whichinlocalcoordinatesisdescribedbyasymmetricmatrix
G(x)={gij(x)},sothatforanytwovectorsv,w∈TxM
hv|wiG= n
X
i,j=1gij(x)viwj=vTG(x)w.
ThemetricdefinesanormofvectorsbykvkG=√hv|viG,and
hencethelengthofcurvesinM:givenacurve joiningpoints x0andx1intime,andsuchthat ˙ (t)=v,thelengthof is
givenbyℓ( )=
R
0kv( (t))kGdt.Thesimplest example ofa Riemannianmanifold isthe standard Euclidean space Rn: the local coordinates are the usual Cartesian coordinates, and the scalar product is defined by the identity matrix G(x)=Id. In particular,
the distance between two points p and q is d(q,p)=
p
(q1−p1)2+···+(qn−pn)2,andmoregenerally,theshortest
pathbetweentwopointsisastraightline.Thesefactsareno moretrueinRiemannianmanifoldswithnon-trivial,i.e., non-Euclidean,metricstructure.Infact,theshortestpathbetween twopointsisthegeodesiccurveofthemetric,i.e.,thecurveof minimallength,likeforinstance,the meridiancircleson a sphere.IntrinsictopologicalpropertiesofRiemannian mani-foldscanbecharacterizedintermsoftheircurvaturetensor, butwewillnotdiscussithere.Inwhatfollows,inordertoavoid anyambiguity,weuse k·kG andh·|·iG todenotethe scalar
productsandnormscomputedwithrespecttothe Riemann-ianmetricG,whilek·kandh·|·iwilldenotetheirEuclidean equivalents.
WeconcludethisshortreviewofRiemanniangeometryby recallingthemeaningofagradientofafunction(Dubrovin et al. (1991)). Let f be a smooth function inthe Riemann-ianmanifoldMequippedwithsomemetricG.Itsdifferential dxf=
P
ni=1∂xif(x)dxi isalinearoperatorinTxM,alsocalleda differential1-form.Definition1. Avectorw∈TxMiscalledthegradientofthe
func-tionfatapointx∈Mifitsscalarproductwithanyothervector v∈TxMisequaltothedirectionalderivativeoffwithrespect
tovcomputedatx:
vf(x)=dxf(v)=vT∇f(x)=hv|wiG.
Intherestofthispaper∇Gf(x)denotesthegradientoffinM
definedinthesenseofthemetricG,i.e., vf(x)=hv|∇Gf(x)iG.
Itiseasytoverifythatinlocalcoordinates∇Gf(x)isrelatedto
theusualEuclideangradient∇f(x)asfollows:
∇Gf(x)=G−1(x)∇f(x). (3)
3.
Open
evaporation
of
homogeneous
multicomponent
mixtures
Letusconsideranopenevaporationprocessofan-component homogeneousmixture.Weassumethattheprocessisisobaric (P=const)andthatthethermodynamicequilibriumbetween liquidandvapourphasesispreserved.
3.1. Thestatespace
For i=1,...,n denotebyxi, yi the partialmole fractionsof
theithcomponentintheliquidandvapourphases respec-tively.Since
P
ni=1xi=1,then−1independentmolefractionsoftheliquidphasex=(x1,...,xn−1)belongtotheGibbssimplex
={xi∈[0,1]:
P
i=1n−1xi≤1}.Inwhatfollows∂willdenotethe boundary of. According tothe Gibbsphase rule, the system underconsideration hasndegreesoffreedom,and itsthermodynamicalstatecanbedescribedintermsofn−1 independentmolefractionsoftheliquidphasexandthe tem-peratureT.So,thestatespaceofthesystemisthedifferential manifoldM={q=(x,T): x∈, T∈R}.
3.2. Partialmassbalanceandboilingtemperature
Inthestandardequilibriummodelofopenevaporationa mul-ticomponentliquidmixtureisvaporizedinastillinsuchaway thatthevapouriscontinuouslyevacuatedfromthecontact withtheliquid(DohertyandPerkins,1978).Thepartialmass balanceofsuchasystemcanbewrittenintheform
dxi
d =xi−yi(x,T)=vi(x,T), i=1,...,n−1. (4)
Here∈[0,1]isanon-decreasing parameterdescribingthe change intheoverallmolarquantityofthe liquidphasenl
intimet:=ln(nl(0)/nl(t)).Solutionstosystemofdifferential
equations(4)arecalledresiduecurves,andtheirgraphical rep-resentation inthe simplexformsthe residue curve map (RCM). Therighthand sideof(4) definesa vector fieldv= (v1,...,vn−1)∈TMcalledtheequilibriumvectorfield. Its
singu-larpoints,i.e.,thepointsq∈Msuchthatv(q)=0describethe purecomponentsandtheazeotropesofagivenmixture.
Bydefinitionofmolefractions,
n
X
i=1
yi(x,T)=1. (5)
Thisconstraintdefinesahyper-surfaceWinthestatespace
M,calledtheboilingtemperaturesurface,whichisinvariantwith respectto(4).Sinceinahomogeneousmixtureeach compo-sitionoftheliquidphasexischaracterizedbyauniquevalue ofT,inprincipleequation(5)canbesolvedinordertoexpress
ofagivenmixture.1Inotherwords,theboilingtemperature surfacecanberepresentedasagraphoffunctionTb:
W={q∈M: q=(x1,...,xn−1,Tb(x1,...,xn−1))}.
It is worth to underline that in practice, due to the high complexityofthermodynamicalmodelsofrealmixtures,the functionTb(x)cannotbewrittenexplicitly.Nevertheless,ifthe
mole fractionsyi(x, T)are known,equations (4), (5) forma
closedsystemofdifferentialalgebraicequations,whichcanbe solvednumerically.Thisisthestandardchemicalengineering approachforpracticalcomputations.
Despiteitspracticalutility,themodelmadeofequations
(4)and(5)isnotsuitableforthequalitativeanalysisofRCM. Inparticular,itgivesnotanswertothequestionsposedinthe Introduction.Forthiswehavetolookcloseratthe thermody-namicequilibriumcondition.
3.3. ThevanderWaals–Storonkinequation
Adifferentwaytoexpressthethermodynamicequilibriumis providedbythevanderWaals–Storonkinequationsofphase co-existence.Theirrigorousmathematicalderivationfromthe equilibriumconditiondG=0,whereGisthetotalGibbsfree energyofthesystem,canbefoundinStoronkin(1967),Zharov and Serafimov (1975) and in the reviewpaper (Toikkaand Jenkins,2002).Theequationfortheliquidphasereads
sv−sl+ n−1
X
i=1 (xi−yi) ∂sl ∂xi!
dT− n−1X
i,j=1 ∂2gl ∂xi∂xj(xi− yi)dxj=0. (6)Heresl and sv are the entropies ofthe liquid and vapour
phases,and gl istheGibbsfreeenergyoftheliquidphase.
ThetermscontainingdPareneglectedin(6)sinceonly iso-baricprocessesareconsidered.Ananalogousequationcanbe alsowrittenforthevapourphase.Inthecasen=1equation
(6)impliestheClausiusequation,whereasifn=2itbecomes theclassicalvanderWaalsequation.
Letusseeunderwhichconditionthemodel(4),(5)basedon themassbalanceargumentsisconsistentwithstateequation
(6).
3.4. TheGibbsmetric
Considerthefollowingdifferential1-forminthestatespace
M: =1s dT+ n−1
X
i,j=1 ∂2gl ∂xi∂xj vidxj, where 1s=sv−sl+ n−1X
i=1 (xi−yi) ∂sl ∂xi = nX
i=1 yi(svi −s l i), sli,svi beingthepartialmolarentropiesoftheithcomponent
ineachphase.InwhatfollowswedenotebyD2
xgl={∂x2ixjgl}n−1 i,j=1
theHessianmatrixofglwithrespecttox
i,i=1,...,n−1.The
materialstabilityconditionimpliesthatD2
xgl definesa
posi-tivedefinitequadraticform,while1s>0forall(x,T)∈M,in particular,ontheboilingtemperaturesurfaceW.
1 ThisfactfollowsfromtheImplicitfunctiontheorem.
Geometricallyspeaking, equation(6) meansthat ifıxis apossibleinfinitesimalchangeinthesystemunder thermo-dynamicequilibrium,then(ıx)=0.Allpossibleinfinitesimal
changes in the system under thermodynamic equilibrium
formavectordistribution6={ıx∈TM:(ıx)=0}.Observethat the form placesrestrictions onthe possibledynamicsof thesystemratherthanonitsstateatagivenmoment,while the boilingtemperaturesurfaceW⊂Mrepresentsall possi-blestatesofthesystemcompatiblewiththethermodynamic equilibrium.ThereforeTW⊂6.SincethetangentspacetoWis spannedbyvectorsoftheformri=∂xi+∂xiTb∂T,weget(ri)=0
fori=1,...,n−1,andhence
∇Tb(x)= 1
1sD
2
xgl|T=Tb(x) v(x), x∈. (7) Thepositivedefinitenessof 1
1sD2xgl|T=Tb(x) inallowsusto introduceaRiemannianmetricinassociatedtothe sym-metricmatrix
Ŵ(x)=1s1 D2xgl|T=Tb(x),
whichwewillcalltheGibbsmetric.Comparisonofformulae(7)
and(3)yields v(x)=∇ŴTb(x).
CalculatingnowthederivativeofTbalonganyresiduecurve
x(),weget dTb(x())
d =v(x())
TŴ(x())v(x())
=kv(x())k2Ŵ. (8)
Remark. Althoughthe1-formisdefinedeverywhereinM, someofthesecondderivatives∂2
xixjglblowupatthepure
com-ponents(verticesof)andontheedgesof(seeexamples later).So,strictlyspeaking,theGibbsmetricŴisawelldefined Riemannianmetriconlyintheinteriorpointsof.
Theabovecomputationscanbesummarizedasfollows. Theorem 1. Theopen setint={xi∈(0,1):
P
n−1i=1xi<1}ofpartialmolefractionsendowedwiththeGibbsmetricŴisa Riemann-ianmanifold.Residuecurvesaresolutionstothegradientsystem
dx
d =∇ŴTb(x), x∈, (9)
wheretheboilingtemperatureTbplaystheroleofthepotential func-tion.Moreover,alonganyresiduecurvex(),theboilingtemperature changesaccordingtotheequation
dTb(x())
d =kv(x())k
2 Ŵ,
andthusitisanaturalLyapunovfunctionforsystem(4).
Remark. Eqs.(8)and(9)providetheexplicitanswersto ques-tionsQ1andQ2formulatedinSection1.Wealsoremarkthat Eqs. (8) and (7) are well known inthe residue curves the-ory(see,forinstance,inZharovandSerafimov,1975,Doherty and Perkins, 1978, vanDongenand Doherty, 1984), though theirintrinsicgradientstructurewasdenied(vanDongenand Doherty,1984;Rev,1992).
LookingatresiduecurvesthroughtheopticoftheGibbs met-ric, onecan deriveall qualitativepropertiesofRCM asthe trivialconsequenceofthegradientformofsystem(4).Indeed,
Fig.1–Idealmixture:methanol(x1),ethanol(x2)and1-propanol.(a)Theboilingtemperaturesurfaceanditsisothermlevel
setsonit;(b)theresiduecurvesmap.
propertiesofRiemanniangradientsystemsarewellknown, andtheyareanalogoustothepropertiesoftheclassical gra-dientsystemsintheEuclideanspaceRnmodulothechangeof themetric(seeforinstanceinHirschetal.(2004)).Inparticular:
– criticalpointsofTbaresingularpointsof(4)inint;
– generically,theycanbestable/unstablenodesorsaddles; – if cisaregularvalueofTb, i.e.,if∇Tb|T−1
b (c)=/0,thenthe vectorfieldvisorthogonaltothelevelsetT−1
b (c)inthesense
oftheGibbsmetricŴ.
The first two properties are well known and widely used inthe RCM analysis.The thirdoneis truewithinthe Rie-mannian viewpoint, but it is not if we use the Euclidean metric.Thisexplainsthedebateaboutisothermsandresidue curvesorthogonalityintheliterature.However,thehigh non-trivialityoftheGibbsmetricŴmakesthetopologyofresidue curvesmapsmuchmoresophisticatedthantheoneofa clas-sicalgradientflow.
One may ask what happens when a residue curve x()
approachestheboundary∂oftheGibbssimplex?Inthis casetheGibbsmetricblowsupandk·kŴisnotdefined,while
kvk→0andgenerically2∇T
b|∂ =/ 0andhasbounded
compo-nents.So,kvkŴ=
p
hŴ−1∇Tb|∇Tbistaysboundedon∂,and
henceasx()approachesacriticalpointx*∈∂(apure
com-ponentor an azeotrope oforder <n), kv(x())k∼kv(x())k2Ŵ as
x()→x*.Observe that thesituation isdifferent ifx*∈int
isanazeotropeofordern,inthiscasekv(x())k∼kv(x())kŴ.
So,theborderandinternalsingularitiesoftheresiduecurves mapsinareofdifferentnature:whileinternalazeotropes arecriticalpointsoftheboilingtemperature,thesingularities atpurecomponentsandatazeotropesoforder<nresultfrom theblowupofthemetricŴ.Moreover,sincetheboiling tem-peratureisnotdecreasingalongresiduecurves,thosecanbe re-parametrizedbytakingTbinsteadof:
dx dTb =
v(x) kv(x)k2Ŵ
. (10)
2 i.e.ifthereisnotangentialazeotropes.
ThistransformationcanbeusedtoregularizethewholeRCM ifitcontainsnointernalazeotropes.Inparticular,thisshould simplifythenumericalintegrationofresiduecurvessincethe newparameterisbounded:Tb∈[Tbmin,Tmaxb ],while∈[0,+∞).
4.
Ternary
mixtures:
first
results
Thefirstnon-trivialsituation,wheretheGibbsmetricappears, concernsthesimpleevaporationofthree-components mix-tures. In addition, inthis case we caneasily visualize the conceptsintroducedabove.TheRiemannianviewpointmakes clearstructuralpropertiesoftheRCM,forinstance,the rela-tion between residue curves and isotherms, and between distillationboundariesandridge/valleycurvesoftheboiling temperaturesurface.
4.1. Residuecurvesandisotherms
Weconsiderhereonlythegenericsituationofhomogeneous ternarymixturewithouttangentialazeotropes.Choosingtwo independentmolefractionsx1,x2,theboilingtemperature
sur-faceWovercanbeseenasa2Dsurfaceina3DEuclidean spacewithcoordinatesx1,x2,T.Onthex-plane,alongwith
the residue curves,we have another family ofcurves, the isotherms,definedastheprojectionsofthelevelsetsT−1
b (c)
oftheboilingtemperature.Atanypointx∈int,thetangent vectortotheisothermisgivenbyw=(−∂x2Tb(x),∂x1Tb(x))=
(∇Tb(x))⊥.TheŴ-orthogonalityofvectorsvandwcanbe
veri-fieddirectly:
hv|wiŴ=hŴv|wi=hŴŴ−1∇Tb|(∇Tb)⊥i=0.
Moreover, away from ternary azeotropes the vectors ev=
v/kvkŴandew=w/kwkŴformawelldefinedŴ-orthonormal
basisinint.
Example 1. The ideal mixture methanol (x1)-ethanol (x2)
-1-propanol.Ourcomputationsarebasedonthe3-suffix Mar-gulesmodelforactivitycoefficients(Prausnitzetal.,1998).In
Fig.1b,weshowtheRCM,whileFig.1ashowstheboiling tem-peraturesurfaceWwherethethincurvescorrespondtothe isothermlines.Anycurve(x(),T(x())onWprojectsonthe
Fig.2–ComponentsofŴ(thickcurves)vs.componentsof
e
Ŵ(dashedcurves). residuecurvex()(boldcurveinFig.1b)ontheplane(x1,x2).Thevectorsvandwdescribedabovearenotorthogonalalong x(),asshowninFig.1a.Onmaythinkthattheirprototypeson
TWareorthogonal.Tocheckthispropertywehavetocompute theirscalarproduct,thusweneedtochooseametriconW.
4.2. 3Dgeometryoftheboilingtemperaturesurface
AlongwiththeGibbsmetricŴ,thereisanothermetriconW, whichdescribesthe embeddingofthegraphoffunctionTb
intothe3DEuclideanspacewithcoordinatesx1,x2,T.Indeed,
assumethatWisendowedwithsomeRiemannianmetric
e
Ŵ. ThelengthofanycurvelyingonWcanbecomputedintwo differentways:asalengthwithrespecttothemetrice
ŴonW, orasalengthofthesamecurveconsideredintheambient3D Euclideanspace.Thereexistsauniquechoiceofe
Ŵthatassures thatthesetwolengthscoincide(Dubrovinetal.,1991).Definition2. TheRiemannianmetricontheboiling tempera-turesurfaceW={q∈R3: q=(x1,x2,Tb(x1,x2))}associatedto
thequadraticformwithcomponents
e
Ŵ11=1+ ∂T b ∂x1 2 ,e
Ŵ22=1+ ∂T b ∂x2 2 ,e
Ŵ12= ∂Tb ∂x1 ∂Tb ∂x2.iscalledthenaturalRiemannianmetric
e
Ŵ.In Differential Geometry the natural metric is also called theI-stfundamentalformofasurface.Roughlyspeaking,it describesthevisibleshapeofthesurfacein3D.Unlikelythe GibbsmetricŴ,thenaturalmetric
e
Ŵiswelldefinedandfinite everywherein,in,particular,onitsboundary.Itisimportant tostressoutthatŴande
ŴdefinetwodifferentgeometriesonW.
Example1(Continuation). InFig.2wecomparethe compo-nentsoftheGibbsmetricsŴwiththecomponents
e
Ŵinthe sectionx1=0.2fortheidealmixtureofFig.1.ObservetheblowupoftheGibbsmetric(thickcurves)intheneighborhoodof theboundaryof.Letusnowcomebacktothelastquestion oftheprevioussubsection.Denoteby()theanglebetween theequilibriumvectorvandthetangentvectortoisothermw alongx().Remarkthatdependsonthechoiceofthemetric
gsincecos=kvkhv|wigkwkgg.Fig.3showsthevariationofthecos alongthetestcurvex()ofFig.1baccordingtothreepossible metrics:theEuclideanmetricg=Id(dashedcurve),the natu-ralmetric
e
Ŵ(thickdashedcurve),andtheGibbsmetricŴ(thick curve).Onlyinthelattercasecos<10−2,whichmeansthatvandwareŴ–orthogonalwithintheaccuracylimitsofthe model.
4.3. Ridge/valleyscurvesoftheboilingtemperature
Aswealreadymentioned,therelationbetweenthedistillation boundariesandridge/valleycurvesontheboilingtemperature surfaceWisstilldebated.Variousdefinitionsforridge/valley curveswereproposedintheliterature(Kivaetal.,2003),but inouropinionnoneofthemissatisfactory.Letusanalyzethe rigorousdefinitionofthisobjectusedinDifferentialGeometry takingintoaccountthenon-trivialRiemannianmetricŴ.
Consider a gradient dynamical system ofform (9), and denotebyCtheisothermcorrespondingtothelevelsetTb=c.
ApointxonCbelongstoaridgeoravalleyofTbifk∇ŴTb(x)k2Ŵ=
kv(x)k2Ŵhasamaximumoraminimumatthispoint(Boscain
etal.(2013)).Thisleadstothefollowingdefinition,which pro-videstheanswertothefirstpartofquestionQ3.
Definition3. Theridge/valleycurvesoftheboiling tempera-turearelociofpointsx∈intsuchthat
h∇Ŵkvk2 (x)|wiŴ=wkvk2Ŵ(x)=0, (11)
where wkvk2
Ŵ denotes the directionalderivative of kvk2Ŵ with
respecttow,wbeinganytangentvectortotheisotherm pass-ingthroughx.
According toDefinition 3, the ridge/valley curves ofthe boilingtemperatureareintrinsicallyrelatedtothe Riemann-iangradientofTb.Eq.(11)impliesthatazeotropesandpure
componentsbelongtoridge/valleycurves.Moreover,theyare tangenttotheeigenvectorsoftheJacobianDxvatazeotropes
and pure components. Observe also that knowingjust the function Tb(x1, x2) over is not enough to compute the
ridge/valleycurves:inadditiononeneedstoknowtheGibbs metric Ŵ. Forthis reasonthe shapeofthesurfaceWin3D cannotbeused todefinethe distillationboundariesasthe
Fig.3–OrthogonalitytestalongtheboldcurveofFig.1in differentmetricson.
Fig.4–Non-idealmixture:benzene,acetone(x1),chloroform(x2).(a)Theboilingtemperaturesurfaceandisothermlevel
sets;(b)theresiduecurvesmap;(c)theheightsurfaceofkvk2
Ŵ;(d)ewkvk2Ŵalongthedistillationboundary.
projectionsitsflexuresassomeotherauthorsdid(vanDongen and Doherty, 1984, Rev, 1992 and other references in Kiva etal.,2003).Infact, itfollowsthatinordertovisualizethe ridge/valleycurvesoftheboilingtemperature,wehaveto ana-lyzethelandscapeoftheheightsurfaceofkv(x)k2Ŵratherthan
theWitself.
Remark. ObservethatifbothTbandŴ(andhencev)areknown
forallx∈,theridge/valleycurvescanbedetectedbyfinding zerosofthe scalartestfunctionwkvk2
Ŵ withoutsolvingany
differential equation,which makesthis notionparticularly interestingfromthecomputationalpointofview.
Thenexttwoexamplesillustrateourconstructionforternary mixtureswithdistillationboundaries.Inbothcasesweused
the thermodynamic model based on the NRTL equations
(Prausnitzetal.,1998).
Example 2. Benzene– acetone (x1)–chloroform (x2)
(Serafi-mov’stopologicalclass1.0−2accordingtoKivaet al.,2003).
ThemainfeaturesofthismixtureareshowninFig.4:ithasa binaryazeotropeofsaddletypeatthepointxaz≈(0.351,0.649),
characterizedbytheboilingtemperatureTaz
b ≈65.11◦C.The
distillationboundaryistheseparatrixcomputedvia numeri-calintegrationofsystem(4)onW.Itisdisplayedbythethick blackcurve,whichstartsatxaz andgoestotheorigin
(ben-zenepurecomponent, Tb≈80.10◦C).InFig.4cweshowthe
heightsurfaceofkvk2Ŵ:thethincurvesaretheisoclines,and
the thick blackcurve showsthe positionofthedistillation boundaryonit.Weseethatthedistillationboundarypasses veryclosetothebottomofthevalleyoftheheightsurface ofkvk2Ŵ.Nevertheless,thecomputationofthefunctionewkvk2Ŵ
alongit(Fig.4d)showsthatitdivergesfromthevalley’s bot-tomwhileapproachingthepointx=(0,0)correspondingtothe benzenepurecomponent.
Example 3. Methanol–acetone (x1)–chloroform (x2)
(Serafi-mov’s topologicalclass3.1-4according toKivaet al.,2003),
Fig.5–Non-idealmixture:methanol,acetone(x1),chloroform(x2).(a)Theboilingtemperaturesurfaceandisothermlevel
sets;(b)theresiduecurvesmap;(c)theheightsurfaceofkvk2
Ŵ;(d)ewkvk2Ŵalongthedistillationboundaries.
shown in Fig. 5a. As we can see, the topological struc-tureofthe RCM (Fig. 5b)ofthis mixtureis morecomplex. Ithas
– three binary azeotropes: 2unstable nodes at the points
B12≈(0.7928,0) andB13≈(0,0.6536),and onestablenode
atB23≈(0.3511,0.6489)withmaximumbolingtemperature
T23
b ≈65.11◦C;
– oneternaryazeotropeofsaddletypeatthepointA≈(0.3676, 0.2107), characterized by the boiling temperature Taz
b ≈
56.99◦C.
Four separatrices (thick black curves) form the distillation boundaries, which divide the RCM into four distillation regions.Asinthepreviousexample,theywerecomputedby numericalintegrationofsystem(4).Fig.5.cshowstheheight surfaceofkvk2Ŵ.Asbefore,thethincurvesareisoclines,and
the thick blackcurves indicate the location of the distilla-tionboundaries.Whilethedistillationboundaryconnecting thepointsB12,B13 seemstofollowthebottomofthevalley
ofthe height surface,the curve connecting B23 with x=(0,
0) (methanol pure component) diverges significantly from the ridge.Infact,this divergencebecomeevidentfrom the computationofthetestfunctionewkvk2Ŵ(Fig.5.d):the thick
curvecorrespondstothecurveconnectingB23with(0,0),the
dashedcurvecorrespondstothecurve connectingB12with
B13. Weconcludethis example byshowingthebehaviorof
thecomponentsofGibbsmetricoverthecompositionspace (Fig. 6).As stated inSection3.4, theyblow up along the boundaryofGibbstriangle,butstayregularattheternary azeotrope.
Inbothoftheaboveexamplesweobservedthatdistillation boundariesdonotcoincidewiththeridge/valleycurvesofthe boilingtemperature.Onemayaskiftheobserveddifference betweenthetwotypesofcurvesisrelatedtotheaccumulation ofthenumericalerror,andDefinition3issuitabletodefine distillationboundaries.Aswewillshowinthenextsection withasimpleacademicexample,theanswerisnegative:in generalthereisnoreasonforthesetwofamiliesofcurvesto coincide.
Fig.6–LevelsurfacesofthecomponentsofGibbsmetricforthemixturemethanol-acetone-chloroform
4.4. Ridge/valleysanddistillationboundaries
Let us now check if the notion of the ridge/valley curves introducedin Definition 3is consistentwith the notionof adistillationboundary.Weprecisethatdistillationboundaries
areseparatrices ofRCMconnecting stable/unstablenodes to saddles.3Inparticular,ifacurve (·)
∈isadistillation bound-ary,then:
(a) itisaresiduecurve,i.e.,itisanintegralcurveofsystemof differentialequations(4),andhenceitcannotbecrossed byanyotherresiduecurve;
(b) itstartsatanunstablenodeandfinishesatasaddle,or startsatasaddleandfinishesatastablenode;
(c) atitsterminalpointsitistangenttotheeigenvectorsof theJacobianDxv,or,equivalently(ZharovandSerafimov
(1975)),oftheHessianD2 xTb.
Aswesaw,properties(b)and(c)arealsoverifiedbyridge/valley curvesdefinedbyequation(11).Thisiswhyitseemsnatural toexpectthatthetwodefinitionsareequivalent.Toconclude, onemusttestwhethertheridge/valleycurvesverifyproperty (a).Letusputthisquestioninamoregeneralmathematical context.Onaplane(x1,x2)consideragradientsystem
asso-ciatedtosomepotentialfunctionF,andassumethatithasat leastonenodeandonesaddle.Isittruethattheridge/valley curveofFdefinedbyequation(11)isananintegralcurveof thegradientsystem ˙x =∇F(x)whateveristhemetricofthe plane?Ingeneral,theanswerisnegative,asitisshownbythe followingacademiccounterexample.
Example4. Forsimplicity, we considerthe Euclidean case whereall the computationscanbedone explicitly. LetF= x1−x31−(x2−x1)2.Thecorrespondinggradientsystemis
˙x1=1−3x21−2(x1−x2), ˙x2=2(x1−x2), (12)
ithasasaddletypesingularityatthepointA=(−1/√3,−1/√3) and a stablenode atB=(1/√3,1/√3).Its phaseportrait is showninFig.7,wherethickblackcurvesrepresentthe sepa-ratriciescomputednumerically.Equation(11)canbewritten intheform
4
#
−9x41−4x1+1+8#
9x31+6x21−3x1+2x2−48x1x22=0,3 Inthetheoryofdynamicalsystemssuchcurvesarecalled
het-eroclinicorbitsofsystem(4).
whichyieldstwofamiliesofridge/valleycurves(reddashed curvesinFig.7): x2= 9x3 1+6x21±
q#
3x2 1−1 2#
9x2 1+4 −3x1+2 12x1 .AsFig.7shows,thedashedcurvesdonotcoincidewiththe blackcurvesrepresentingtheseparactrices.Inordertoavoid anydoubtconcerningthenumericalerrorrelatedtothe inte-grationof(12),observethatthepointx0=(0,−1/4)belongsto
theridgecurve,whichatthispointistangenttothevector(16, 13).Ontheotherhand,∇F(x0)=(1/2,1/2),sotheridge/valley
curvepassingthroughx0isnotasolutionto(12),andhenceit
canbecrossedbyintegralcurvesof(12).
NowweareabletocompletetheanswertoquestionQ3 posedintheIntroduction:theridge/valleycurvesofDefinition 3connectthesingularpointsofRCMandaretangenttothe distillationboundariesatthesepoints,but ingeneral,they are notresiduecurves,andthustheycannotbedistillation boundaries.So,althoughweareusingadifferentdefinitionof
Fig.7–Phaseportraitofthegradientflowof F=x1−x3
1−(x2−x1)2:separatrices(thickblackcurves)and
ridge/valleycurves,weconfirmthegeneralconclusionmade byvanDongenandDoherty(1984)andbyotherauthors(Rev, 1992;Kivaetal.,2003).
5.
Conclusion
Thethermodynamicalequilibriumcondition,describedbythe vanderWaals–Storonkinequation (6)endowsRCMofopen evaporationby a non-trivialRiemannianmetric: the Gibbs metricŴ.Withinthisgeometricalmodel,anyRCMistheset oftheintegralcurvesoftheRiemanniangradientflow asso-ciatedtotheboilingtemperatureTb,whichplaystheroleof
apotentialfunctionforRCM.Thiskeyfactexplainsall well-knownpropertiesoftheresiduecurvesmapsandoftherole ofTb,andinaddition,itimpliesthatunlikelyintheclassical
Euclideancase,theequilibriumvectorfieldisŴ-orthogonalto theisothermfrontsawayfromazeotropes.
Weexploredthefirstnon-trivialconsequencesofthis geo-metricviewpointinthecaseofternarymixtures.Inparticular, wediscussedtherelationbetweendistillationboundariesand the ridge/valley curves of the boiling temperature Tb and
showed that they do not coincide ingeneral since the Tb
ridge/valleycurvesarenotresiduecurves.
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