On the Riemannian structure of the residue curves maps

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T

OULOUSE

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residue curves maps. (2015) Chemical Engineering Research and

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On

the

Riemannian

structure

of

the

residue

curves

maps

Nataliya

Shcherbakova

a,∗

,

Vincent

Gerbaud

b

,

Ivonne

Rodriguez-Donis

b

a_{LaboratoiredeGénieChimique,UniversitéPaulSabatier–INPENSIACET,4,alléeEmileMonso,CS84234,31432}

Toulouse,France

b_{LaboratoiredeGénieChimique,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:[email protected](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,whilegl_istheGibbs

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 other words,thereisapositivedefinitequadraticformcalledmetric, whichinlocalcoordinatesisdescribedbyasymmetricmatrix

G(x)={gij(x)},sothatforanytwovectorsv,w∈TxM

hv|wiG= n

X

i,j=1

gij(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

n_i=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

n_i=1xi=1,then−1independentmolefractions

oftheliquidphasex=(x1,...,xn−1)belongtotheGibbssimplex

={xi∈[0,1]:

P

_i=1n−1xi≤1}.Inwhatfollows∂willdenote

the 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.1_{Inotherwords,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−1

X

i,j=1 ∂2_gl ∂xi∂xj(xi− yi)dxj=0. (6)

Heresl _{and s}v _{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 ∂2_gl ∂xi∂xj vidxj, where 1s=sv−sl+ n−1

X

i=1 (xi−yi) ∂sl ∂xi = n

X

i=1 yi(svi −s l i), sl

i,svi beingthepartialmolarentropiesoftheithcomponent

ineachphase.InwhatfollowswedenotebyD2

xgl={∂x2_ix_jgl}n−1 i,j=1

theHessianmatrixofgl_{withrespecttox}

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=∂x_i+∂x_iTb∂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())

T_{Ŵ(x())v(x())}

=kv(x())k2Ŵ. (8)

Remark. Althoughthe1-formisdefinedeverywhereinM, someofthesecondderivatives∂2

x_ix_jglblowupatthepure

com-ponents(verticesof)andontheedgesof(seeexamples later).So,strictlyspeaking,theGibbsmetricŴisawelldefined Riemannianmetriconlyintheinteriorpointsof.

Theabovecomputationscanbesummarizedasfollows.

Theorem 1. Theopen setint_={xi∈(0,1):

P

n−1_i=1xi<1}of

partialmolefractionsendowedwiththeGibbsmetricŴisa 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:asalengthwithrespecttothemetric

e

ŴonW, orasalengthofthesamecurveconsideredintheambient3D Euclideanspace.Thereexistsauniquechoiceof

e

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 tostressoutthatŴand

e

Ŵdefinetwodifferentgeometrieson

W.

Example1(Continuation). InFig.2wecomparethe compo-nentsoftheGibbsmetricsŴwiththecomponents

e

Ŵinthe sectionx1=0.2fortheidealmixtureofFig.1.Observetheblow

upoftheGibbsmetric(thickcurves)intheneighborhoodof theboundaryof.Letusnowcomebacktothelastquestion oftheprevioussubsection.Denoteby()theanglebetween theequilibriumvectorvandthetangentvectortoisothermw

alongx().Remarkthatdependsonthechoiceofthemetric

gsincecos=_kvkhv|wi_gkwkg_g.Fig.3showsthevariationofthecos alongthetestcurvex()ofFig.1baccordingtothreepossible metrics:theEuclideanmetricg=Id(dashedcurve),the natu-ralmetric

e

Ŵ(thickdashedcurve),andtheGibbsmetricŴ(thick curve).Onlyinthelattercasecos<10−2_{,whichmeansthat}

vandwareŴ–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.3_{Inparticular,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+2

x2−48x1x22=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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