ON THE DYNAMICS OF DISTILLATION PROCESSES-III

THE TOPOLOGICAL STRUCTURE OF TERNARY RESIDUE CURVE MAPS

M F DOHERTY Department of Chemical Engmeenng. Goessmann Laboratory. Umverslty of Massachusetts, Amherst, MA 01003,

USA

and

J D PERKINS Department of Chenucal Engmeenng and Chemical Technology, Impenal College, South Kensmgton, London SW7

2BY. England

(Received for pubhcatwn 6 Apnl 1979)

Abstract-The dtierentlal equations descnkung the sunple dlstdlatlon of azeotropic ternary mixtures place a physrcally meanmgful structure (tangent vector field) on ternary phase &grams By recogmzmg that such structures are subJect ot the PorncarLHopf mdex theorem It has been possible to obtam a topologrcal relatlonshrp between the azeotropes and pure components occumng m a ternary mature Thus relahonshlp gives useful mformatlon about the dlstdlatlon behavior of ternary rruxtures and also predicts sltuahons m which ternary azeotropes cannot occur

INTRODUCTION

One of the most tnterestlng problems m the ther- modynanucs of azeotrop~c rmxtures IS the theoretical predIction of the hkehhood of the appearance of ternary azeotropes usmg only a knowledge of the propeties of the three constituent bmary mrxtures If a detailed knowledge of the bmary mixtures IS avadable (as expressed by knowmg the parameters m a hqrud soluUon model for each of the bmary paus) then It 1s possible to determme the presence or absence of a ternary azeo- trope by numerically solvmg the azeotroplc equlhbnum equations

Px, -P:(T)x,yi(A,x)=O I = 1,2,3 (1)

In wntmg these equations we have assumed that the vapour LS an adeal mixture and that the ternary actwlty coefficients can be determmed from a knowledge of the binary mteraction parameters

Unless we are mterested m the detied behavlour of a spectic system this approach requires too much m- formation and provides httle InsIght as to why and when ternary azeotropes occur Furthermore, there 1s no guarantee that the model even gives quahtahvely correct results (I e It may predict the presence of a ternary azeotrope which does not exist m reality) It IS known that hquad solution models sometunes do predict quahta- tively incorrect phase behavlour [ 1, 51

The estlmatlon of the concentration regon to which a

thllxtures showmg the behavior gwen m Figs 1, 10 and I I are hsted m Table 1

ternary azeotrope must be confined has been the subject of several arttcles tn the Russian hterature[2-41 (other Refs are gwen tn those works) These articles state necessary condltlons for the occurrence of ternary azeo- tropes As Maruuchev[3] points out, “it IS possible with the aid of the Margules-Wohl method, without calculat- mg hqmd-vapour eqmhbnum data, to estunate the con- centration re@on of the ternary azeotrope from data on the binary systems and pure components The questlon of the existence of the ternary azeotrope at a gven temperature remams open”

It IS possible to obtam a few general results on this question without recourse to solution models In obtam- mg these results we also obtam useful mformatlon about the distlllatlon characteristics of the ternary mixture This will become apparent as the methodology IS des- cnbed

Instead of dealing with solution models, we formulate the pair of ordinary ddferentlal equations

dxa dS=X’-Y’ dx2

dS =x2- y2

and mvestlgate the properties of the smgular pomts (steady state solutions) Clearly, this 1s an enmely equivalent problem but appears to be more amenable to analysts than other formulations

EquaUon (2) defines the locus of the residue hquld compos%tion as It changes with time m a simple dls- tdlation process The tra]ectone.s of (2) are usually called residue curves and typical residue curve diagrams are shown 111 Fig 1 t It can be proved[6-81 that eqn (2) has the followmg properties

1401

1402 M F DOHERTY and J D PERKINS

B (al

C B (bl

C

AL.I!!kL Ah B

lc) C 0

Cd) C

Rg I Examples of resbdue curve maps See Table 1 for example mixtures

P(l) The solutions are confined to he either on the boundary of or interior to the nght mangle of umt side

P(u) The Independent vmable, 5, IS a dlmenslontess measure of time It 1s a nonhnear, stictly monotomcally mcreasmg function of tune and IS defined on the mterval

[O, + 001 P(m) The singular points of eqn (2) occur at all pure

component vertices, bmary azeotropes and ternary azeotropes These points are always isolated

P(W) When the equatkon IS Imeamed about azeotro- plc smgular pomts the elgenvalues are datmct and real It follows that there can be at most one zero elgenvalue

P(v) When the equation 1s hneanzed about pure component vertices the elgenvalues are real but not necessanly dlstmct One zero eigenvalue occurs for every dIrectIon, k, In which

PX = 0

the denvatwe hemg evaluated at the vertex of Interest Therefore, one zero elgenvalue occurs for every binary nurture which exhibits a tangential azeotrope at the vertex of Interest It IS highly unhkely that more than one tangential azeotrope would occur at a vertex at any one time so we will assume that in general, there will be at most one zero elgenvalue More details are gwen m Appendix I

P(vi) Limit cycles cannot occur (vu) The temperature surface IS a naturally occurring

Llapounov function for this equation The movement of the hqmd composltlon, x(t), IS always m a direction which makes the temperature increase

Having estabhshed the general propertles of the solu- tions of the dlfferentml equation we are now m a position to ask the questlon, what IS the relabonshlp between the number and the type of singular points which eqn (2) can display7

Since the tune of Porncar& a large body of mathema- tlcal literature has been devoted to answenng Just such a question for abstract dtierentmi equations The relevant theory is calIed degree theory and by its apphcation we are able to answer the question we have posed regardmg

eqn (2) In order that the non-specmhst reader may follow our

reasoning, we include a short summary of the pnnclpal concepts of degree theory

Degree The followmg IS based on the excelIent monograph by

Temme [9] Let fi be an open set 111 R”, its boundary IS denoted by

Xl and Its closure by fi The degree of a smooth function f fi+ R” Hrlth respect to the regon a at the pomt p E R” -f(&k) -f(Z) IS denoted by deg (f, fi, p) and defined by

deg (fi a, P) = 7 sgn I;(K) (3)

where xi E f-‘(p) n n (I e it IS a solution of the equa- tion f(x) = p) and the sum in eqn (3) IS taken over all such solutions contamed m n The symbol J,(x*) sIgndies the determinant of the Jacobian matnx of first partial denvatlves of f evaluated at the point x, The symbol f(z) represents the image of the set of all points in fl for which J,(x) = 0

It can be seen at the outset that degree theory concerns itself vvlth the propertIes of the solutions of an algebrac equation of the form f(x) = p If we have a set of dlfferentlal equations x = f(x) then degree theory can help us determme the propeties of the singular points d we mvestlgate the degree at p = 0, deg cf. a, 0)

If &(x,) = 0 then the degree of f at p IS defined by

deg ti a, PI = da (f, Q, 4) (4)

where q IS any point near p at which Jf(x,) # 0 for all x, which satisfy f(x) = q The degree deg (f, il. q) is then found by the use of eqn (3) It can be proved that the degree at all pomts 4 near p IS the same and so eqn (4) IS

well defined For two dlmenslonal systems there are two

(equivalent) geometnc ways of finding the degree These

are, C(i) The degree IS the number of hmes that the Image

f(an) encircles p where anticlockwlse rotation 1s coun- ted posltlvely and clockwIse rotation negatively Exam- ples are aven m Fu 2

C(u)

deg (f, Q. PI = g (3

where At? IS the change m angle that f makes with respect to some fixed duecuon as x traverses an in the positwe direction (antlctockwse)

On the dynamics of dlshIlafion processes-III 1403

f (ad

Q-f-&a 609 -0

1 n 1_._qY-&L-J I I

d”

F@ 2 Examples of degree

A necessary (but not sticlent) wn&Uon for the degree to change 1s that p moves across the unage of ail (I e across f(8l-i)) It IS because of thus fact that the degree of a pomt p m the unage of ail (1 e m f(an)) IS not defined If such is the case then much of the mate& described m this section ceases to be apphcable We wll take up this pomt agam later

Degree has many useful propertres, some of them are hsted below

D(1) If f-‘(p) IS empty then deg (f. ik p) = 0 D(2) If deg (f, a, p) # 0 then f(x) = p has at least one

solution m R D(3) Let f,, and f1 be smooth functions from fi+R”

Define for 0 5 t 5 1, ft = rfl + (I- t)fo Suppose that p +Z fi(ail) for all t E 10.11 then deg(ft,il,p) IS m- dependent of t In partuzular, deg (f, , fk P) = deg(f,,n,p) Thus 1s a resticted version of the homo- topy mvarumce character of degree and IS a very useful

property D(4) Let p E f(Xk) and K be a closed subset of fi

such that P G f(K) Then deg ti Q, P) = deg (f, i-i - K, p) This 1s called the exclsslon property

We can use degree to define the mdex of an isolated solution of f(x) = p

Index Let f be a smooth function from ai-, R” If x E n IS

an Isolated pomt of f-‘(p) then we define the mdex of f relative to p at the pomt x by,

md Cf, x. P) = deg (f, &(x), P) (6)

where B,(x) IS an open ball of radius I around x such that B,(x) C l2 and contams a smgle solution of the equation f(x) = p Property D(4) ensures that this defimtlon does not depend on r In fact D(4) allows us to replace B,(x) m the above defimtion by any open set around x provided the set hes wholly 111 a and contams only one solution of f(x) = p

It follows from eqn (3) that

md(f,x,p)=sguJ,(x)=sgn G A~ i-1 Q

where A, are the elgenvalues of the Jacobian matnx of f at x (multiple elgenvalues being counted multiple times)

Therefore, when J,(x) # 0 , md (f, x, p) = + 1 or - 1 If Jr(x) = 0 the right hand side of (6) must be replaced by deg(f, B,(x), q) and now the possibdlty exists that f(x) = q has multiple solutions mslde B,(x) In this case deg cf. E,(x). q) may be dtierent from + 1 or - 1. con- sequently md (f, x, p) may be ddferent from + 1 or - 1

By apphcatlon of D(4) we have the useful relatlonshlp,

deg Cf. Cl, P) = z md(f. .G, P) 1

x,Ef’t.P)nn (8)

Consider now the pau of ddferentlal equations,

Assume (9) has k isolated smgular points, xt, xll, xX Then the mdex of the rth smgular pomt, md (f, xp, 0), IS snnply the sign of &(xF) Eqtnvalently, we could draw a small c&e around x;” and note the change 111 angle, A@, which the trajectones crossrng the cucle make relative to some fixed drrection as the circle IS traversed once m the antlclockwlse duectton, then

md (f, xl+, 0) = g (10)

This procedure uses the fact that the tralectory through any point, x, pomts m the same direction as f at that same point

If If(x:) = 0 then the md (f, x,*, 0) cannot be equated with sgn Jf(x:) but the index can stdl be calculated usmg

eqn (10) The geometnc method of calculatmg index 1s therefore

apphcable to both elementary (I e J”(x) # 0) and non- elementary (.&(x) = 0) singular pomts

Bentison ([lo] Chapter X) has summarized this geometnc method very precisely m the followmg theorem

Theorem (Ben&son) Let (0,O) be an Isolated smgular pomt of the dIfferenti

eqn (9) (thus can always be achieved by means of a linear transformation) Either (0,O) IS

(I) a centre or (n) a focus or (m) the open nelghbourhood of the 0~ can be

drvlded mto a fimte number of sectors such that each sector IS either a fan, a hy-perbohc sector or an elhptic sector (see FQg 3) Furthermore, the mdex of the singular pomt at the oqgn IS gven by

md(f,O,O)= l+v (11)

On the dynamics of dlstlllatcon processes-III 1405

lemm and Pollack[l4] Chapter 3 and Mrlnor[lS] Chap- ters 5 and 6

Havmg estabhshed the mathematrcal framework we now return to our problem of studytng the smguhu pomts of eqn (2) subject to the facts P(t)-P(vn) Let us call the mtertor of the rtght trtangle wtth umt stde @, its boun- dary a@ and tts closure & The natural startmg pomt IS to see what f(4) looks hke The fun&tons fr = x1 - yr and f2 = x2 - y2 are constramed to he m the unrt square about the ortgm The Image of &D ts gtven by,

And so we tind the usage space as shown m Ftg 5 We cannot apply eqn (8) for two reasons, smgular pomts of the drtTerentta1 equatrons he on a@ (pure component verttces and bmary axeotropes) and the pomt p = 0 hes on f(M) Rather than attemptmg to find stumble exten- sions of f we well dtscard the classrcal approach m favour of our earlier suggeshon of “piecmg together” eight tnangles to form a closed, hollow 2-dunenslonal polyhedron rn R’ (Frg 6) whtch IS homeomorphtc to the 2-sphere We then tnvoke the Pomcar6-Hopf Theorem m the form,

7 md (f, x,, 0) = 2 (14)

All that remams to be done 1s estabhsh the mdex of the various kmds of smgular pomts of eqn (2). multrply the number of each type occurrmg by its index and sum them to obtam the left hand side of (14) The resuhmg expression 1s an exphctt relattonshtp between the number of each type of smgular pomt occurrmg and every three component axeotroptc mtxture must sattsfy this relatton

The remamder of the paper ts split mto two parts In Part A we assume that all the singular pomts are ele- mentary and in Part B we allow for the posstbthty of

Fig 6 Construction of the polyhedron by “puxmg together” eight compos&on tnangles The polyhedron IS homeomorphlc to

the 2-sphere

non-elementary smguhu pomts tn accordance with pro- perttes P(w) and P(v)

PART A. ELEMENTARY SINGULAR WINTS

For this case there are two kmds of mtenor smguIar pomts shown m Frg 7(a),(b) (ternary axeotropes), two kmds of vertex smgular pomts shown m Ftg 7(c),(d) (pure components) and two kmds of edge smguku pomts shown m Frg 7(e). (f) (bmary azeotropes)

There IS a need to discuss the shape of the trajectones m the vlcrruty of a pure component vertex m more detzul m order to ~ustiy that Figs 7(c) and (d) are the only elementary shapes This 1s done m Appendtces I and II

Let us call Fw 7(d) a pure component quarter saddle and Ftg 7(e) a bmary half saddle Define,

N3 = number of ternary nodes 111 the tnangle S, = number of ternary saddles m the trtangle A% = number of bmary nodes m the trtangle S2 = number of bmary half saddles m the &tangle N1 = number of pure component nodes m the b-tangle S1 = number of pure component quarter saddles m the

trtangle

Frg 5 Compostnon tnangle sad rts image under f

1406 MF

Then from eqn (14).

IN3(8N3)+ r~8s3)+~N2(4N2)+~9(4s2) + IN, (2N, ) + Is, (2Sl) = 2

Do~m-r and J D PERKINS

A (101-c)

(15)

where lN3 IS the mdex associated with a single ternary node on the surface of the sphere, etc The reason for the terms 8N3 and 8S3 m (15) IS that each ternary smgular pomt m the truingle gves nse to e&t unages of Itself on the surface of the sphere Sumlarly, each bmary smgular pomt m the triangle gves nse to four unages of Itself on the surface of the sphere and each pure com- ponent smgular pomt in the mangle gves nse to two unages of Itself

Equation (15) must be coupled w&h one constramt equation,

N,+S,=3 (16)

expressmg the fact that there are exactly three pure components present

The mdlces m (15) take the values,

IN, = + 1 lg = - 1

I ,=+1 r*=-1 (17)

IN‘ = + 1 b, = - 1

The values for 1~3, I-, 1~~ and lNI need no explanation The values for I~ and lsl are both - 1 because these types of singular point m the mangle give nse to genutne four-hyperbohc-sector saddles on the surface of the sphere

Between eqns (U&(17) we obtam the desved refa- tlonshlp.

2N,-2S3+N2-Sz+N,=2 (18) All the diagrams grven m FQ 1 can be seen to conform with thts equation One of the most comphcated ternary

IO) (bl

IO) (I)

Fig 7 Elementary smgutar pomts In the composition tnangle

(7s 9-a

(eo 2%) (60 I-c)

Fa 8 Residue curve map for the system methyl cyclohexane(A) + hexafluorobenzene(E) + benzene(C) at 740 mm

of Hg

diagrams known IS shown m FQ 8 (see Wade and TayIor [16]) and this too conforms vvlth the equation

Equation (18) can be used to enumerate the possible residue curve maps ansmg from any pven set of azeo- tropic data as demonstrated tn the followtng examples

Example 1 Thermodynanucists have often asked whether It IS

possible for a ternary mixture to exhlbtt a ternary azeo- trope m the absence of any binary azeotropes bemg present 1171 The problem IS specfied m Fig 9 It can be seen that vertex 2 IS a quarter saddle and vertices 1 and 3 are nodes, hence, N1 = 2, SZ = 0, N2 = 0 Equation (18) reduces to,

N3-S,=O (19)

which IS satisfied by N3 = S3 = 0,l. 2 but IS not

satisfied by a smgle ternary azeotrope We conclude that when all the smgular pomts are

elementary it is not posstble for a si@e ternary azeo-

T, > Ta > T,

F@ 9 Boundary data for ternary mtxture wth no bmary azeo- tropes

On the dynamics of dlstdlation processes-III 1407

trope to occur in a nuxture which displays no bmary az.eotropes

Example 2 Consider a ternary nuxture with two binary maxunum

bo&ng azeotropes and one bmary muumum boihng azeotrope as shown m Fu lo(a) Vetices A and B are quarter saddles and vertex C IS a node, hence N, = 1 and S, = 2 Also, for this example,

N*+s2=3 (20)

so eqn (18) becomes

Ns-&=2- N2 (21)

Smce N2 IS restncted to take one of the values 0, 1,2,3 we find the followmg structural relationshIps

(I) N2 = 0 implies Ns-S3=2 (22)

(a)

A 1, (100)

F@ 10 (a) Boundary date for a ternary mlxtve w&h three maximum bodmtt bmary azeotropes aad one rmmmum botbng hnary azeotrope (b)-(g) Elementary residue carve maps See

Table 1 for example mutures

The sunplest solution, N, = 2, S, = 0 1s shown m Fig

lo(b) (II) N2 = 1 unphes

N,-S,= 1 (23)

The sunplest solution, N, = 1, S, = 0 is shown m Fig 10(c) (Na = nummum borlmg) and Fig lo(d) (N3 = maxl- mum boding)

(m) N2 = 2 Implies

N3-S,=O (24)

The solution N, = S, = 0 is shown in Fig 10(e) (IV) N2 = 3 implies

N3=Ss-1+N3-Sa=-1 (25)

The simplest solution, N3 = 0, S, = 1 1s shown m Fig 10(f) It 1s mterestmg to note that m Figs 10(b)-(e) there IS only one dlstdlation repon From Fig 10(f) we get the adddlonal information that the bollmg temperature of the ternary azeotrope must be m the interval T4< TAE < T3 A ternary azeotrope cannot exist m the temperature range Ts<TH<T4

There are many ways of organumg the sequence of temperatures m Fx 10 wNe preservmg the specdied boundary condlttons of two bmary maxunum boding azeotropes and one bmary nummum boding azeotrope For example, putting T3 equal to 105 leaves the boundary conditions unchanged but gwes the azeotrope at D a higher bothng pomt than pure A The temperature sequence becomes Tl > Ts > T2 > T. > TS > Ta It 1s

natural to speculate whether the residue curve maps are &ected by rearrangements m the temperature sequenc- mg of this sort Unfortunately, the answer IS yes but the aBect IS not so serious as to destroy the ortgmal structure altogether

Equations (21)-(25) are mdependent of the detads of the temperature sequence, they are based only on the boundary condltlon that there are two maxunum bollmg binary azeotropes and one binary nununum These equations are mvarmnt under boundary condltlon preserving temperature sequencmg rearrangements- This IS sticlent to guarantee preservation of the global fea- tures of the residue curve maps but not sufficient to always guarantee local preservation For example, m the case N3 = S, = 0, we can be sure that N2= 2 for all boundary condition preservmg temperature sequencmg rearrangements (so the global feature IS preserved) but we cannot guarantee that the same two binary azeo- tropes will remam as nodes after the temperature rear- rangement has occurred This can be demonstrated by lettmg T3 = 103, T5 = 101 and T6 = 95 thus preserving the boundary con&tion The new temperature sequence 1s Tt > T3 > T5 > Tz > T6 > T4 Figures lo(b), (c), (d), (f) are preserved both globally and locally under this rear- rangement whde Ftg lo(e) changes to lo(g) Comparmg Figs 10(e) and (g) we see that N2 = 2 for both of them but azeotrope E changes its character, becommg a saddle and azeotrope D becomes a node This change

1408 M F DOHERTY andJ D PERKINS

gves nse to two dlstmct dlstdlation regons m FIN 10(g) whrch contrasts wrth the smgle dlstdlation region m Fig 10(e) So, the temperature rearrangement leads to no change m four of the diagrams but a very dramatic change m the fifth

It IS worthwhrle pomtmg out that the first temperature rearrangement we suggested (puthng T3 = 105) Ieaves aIl the dmgrams unchanged

Thrs kmd of pattern recurs throughout all the examples we have considered which suggests that there IS some simple undertymg rule which governs whether a tem- perature rearrangement alters the residue curve map or not Work IS contmumg m an effort to make this obser- vation quantitatively precise m the hope that qmck screenmg procedures can be devised for the selection of extractrve and azeotroplc dlstiation agents

By reversing all the arrows (I e reversmg tnne) m Figs IO(a)-(g) we obtam the set of diagrams generated by the boundary condltron of two munmum boding bmary azeotropes and one maximum bollmg bmary azeotrope Consequently, we need not consider this as a separate case

Example 3 Consider a ternary murture with three binary maxi-

mum boding azeotropes as shown m Fig 11(a) AU three vertices are nodes so eqn (18) becomes,

N,-S,= l-N* (26)

A T. CIDO)

Fig I1 (a) Boundary data for a ternary muture with three maxlmum bodmg bmary azeotropes (b)-(e) Elementary residue

curve maps See Table 1 for example mixtures

Figure Number

Table 1 Example mixtures for figures shown in text

Component A Component B Reference (Boiling Polnt ‘C) (Boiling Point ‘C)

Caaponent C (Boiling Polnt l C)

1 (a) Acetone (56 4)

1 (b) Benzene (80 1)

1 (cl 1 (d) 1 (d)

lo(b) 10(c) 10(d) 10(e)

10(f)

10(g) 11(b)

11(c)

Aniline (184 4)

Ethylene (-104)

Dichloronethane (41 5)

Iso~ro~l Ether

Ethyl Fotmate (54 1)

Cycl ohexane (80 75)

Paraffin (110)

Acetone (56 15)

Dichlorolnethane (41 5)

Acetone (56.4)

Water (100)

Acetylene (-84)

Methanol (64.7)

No example known

See

See

See

coarsen t be1 ow***

carment belowC”*

Acetone (56 4)

P-Bmmopropane (59 4)

c-n t be1 ow***

Methanol (64 7)

Methanol (64 7)

Carbon Disulphide (46 25)

11 (d) 11 (el

See conenznt below***

No example known

Chloroform (61.2)

Chloroform (61 2)

Hydrazine (113 5)

Ethane (-88)

Acetone (56.4)

Chloroform (61.2)

Chlorofonn (61 2)

Acetone (56.4)

Toluene (110.7)

Methanol (64 7)

Ewe11 and Welch*(20)

Ewe11 and We1 ch*(20) Reinders and BeHinjer

(21.22.23)

Wilson, et al *(24)

Rowlinson(25) (p 184)

Ewe11 and Welch*(20)

Ewe11 and Welch*(PO)

Horsley(26) System numbers 16301. 1450. 5485, 1447

Horsley”(26) System numbers 16327. 5378. 1963. 2079

Benedict and Rubin (27)

Horsley)*(26) System nunbers 16267, 1194. 1175. 1963.

*These papers are concerned with batch drshllatron but contam enough mformatmn to construct sunple dlsUllation resulue curve maps **These systems are topolo@cally slmdar to the dlagmms but ~tb the arrows reversed ***There IS nothmg topolog~cally unusual about these dmgrams although example systems are not readily avmlable m the bterature

On the dynamics of dutdl8hon processes-111 1409

Substitutmg the values 0, 1, 2, 3 for N2 gwes the cases, (1) N2 = 0 unphes

N,-s,= 1 (n)

The sunplest solution, Na = 1, S3 = 0 1s shown m FU 11(b) for the case of a ternary maxunum boll~ng azeo- trope It is clearly ImposslMe for a smgle ternary muu- mum bodmg azeotrope to occur because this reqmres that azeotrope E be a stable node (highest bodmg pomt on the dmgram) which 1s contrary to the statement that Nz=O

(u) N2 = 1 lmpbes

N,-&=O m The sunplest soluaon, N, = Sa = 0 IS shown m Fig 1 l(c)

(m) N2 = 2 Imphes

N,-&=-I (29)

The solution Ns = 0, SX = I 1s shown III Fg 1 l(d) The ternary azeotrope must lie in the temperature range 100<TU<105

(IV) N2 = 3 unphes

N,-s,=-2 m The solution N3 = 0, & = 2 1s shown m Fig 1 l(e)

These !igures appear to be mvammt Consider, for example, the temperature rearrangement T.= 115 with all other temperatures bemg kept the same Figures 1 l(b) and (e) remam exactly the same Fiiures 1 l(c) and (d) are changed m order to accommodate the fact that azeotrope D 1s now the west b&ng pomt but a simple relabel- hng of the axes enables us to recover the form of the o~lgmal diagrams exactly

By reversmg the arrows m these figures we obtam the set of dmgrams generated by the boundary con&Uon of three mmunum bow bmary azeotropes Consequently we need not consider this as a separate case Apply- mg the same reasonmg as m case (1) above we conclude that a smgle ternary maxunum bolllng azeotrope cannot occur when each of the three bmary faces exlublts a mmunum bodmg azeotrope

We could consider examples of ternary nuxtures which exhrbrt fewer than three bmary azeotropes but it 1s not our purpose to present such an exhaustive analysis at ttis pomt

Theorem (Ben-son) The local phase potit of an isolated smgular pomt

with a smgle zero elgenvalue Is of one of the followmg three types node, saddle pomt (four separatnces), two hyperbolic sectors and a fan (three separatices, see Frg 4(d)) The correspondmg m&ces are 1, - 1,O which serve to distmgmsh the three types

This theorem guarantees that the elementary singular points shown m Fig 7 need only be augmented by a few addltlonal types

The zero eigenvalue mtroduces one new type of ter- nary smgular pomt shown m Fig 12(a), two new types of pure component singular pomts shown m Figs 12(b) and (c) and one new type of binary smgular pomt shown m Fig 12(d) The ten dmgrams shown m FQS 7 and 12 exhaust all the possible singular pomts which can occur in a ternary nuxture

We can now denve the more general version of eqn (18) Let us call Fe 12(a) a ternary node-saddle, FQ 12(b) a pure component node-saddle, Fe 12(c) a pure component half saddle and Fig 12(d) a bmary node saddle Define,

NS, = number of ternary node-saddles m the mangle N& = number of bmary node-saddles m the tmu&e N.9, = number of pure component node-saddles m the

tnangle

Then, from eqn (14)

S? = number of pure component half saddles m the *angle

IN,(SN,) + 1&3S,) + lMJ8NS3)

+ 1&4Nz) + 1*(4&) + #~9(4NS2) +I,,(~N,)+I~,(~SI)+IN~,(~NSI)+~~~(~S~)=~ 131)

with the constramt,

Nl+S,+NS,+S:=3 (32)

PART B NON-- ARYBtNGInARPo~

In accordance ~rlth propeNes P(lv) and P(v) of eqn (2) we will now allow for the case of a smgle zero elgenvalue of the Jacobtan matnx

In general, nonelementary smgular pomts are capable of assunung the most mhrcate patterns m the phase plane as a browse through Sansone and ConttC191 WrIl tesbfy However, we are fortunate msofar as we can ICl td) prove that there 1s at most one zero eIgenvahUZ for which pj 12 N case Bendtxson ([lo] p 230) has proved

on-elementary smgular potnts 111 the cornposItion tn-

1410 M F DOHERTY andJ D PERKINS

The rndlces take on the values grven by eqn (17) and usmg eqn (11) we obtam.

lNsj=o INS, = - 1

lN*=o 1&q= -3

Between eqns (17) and (31)-(33) we obtam

(33)

2N,-2&+Nz-Sz+N,-S:=2 04)

Puttmg S? = 0 we obtam the spectal case of eqn (18) We ~111 bnefly review the conclusions given m exam-

ples one, two and three m the l&t of eqn (34)

Example I Our earher conclusion must be mod&d to accom-

modate FM 13(a) which exphcltly demonstrates the way m which a non-elementary pure component smgular point at T, allows for the appearance of a maxnnum bothng (or mmtmum bollmg) ternary azeotrope Ftgure 13(a) conforms to eqn (34) However, it IS nnposslble for a smgle ternary saddle point to occur as can be proved by contradiction

Smce vertices of this sort have not been reported expenmentally we will not pursue this classticafion any further

Example 3 AH we will say here 1s that the conclusion reached

from eqn (27) regardmg the unposslblllty of a ternary muumum botlmg azeotrope ceases to be true when we allow for non-elementary singular pomts This is shown 111 Fui 13(b)

Our analysis of non-elementary singular points has unphcabons about the shape of the boding temperature surface in the viclIuty of ternary azeotropes This 1s discussed m the foliowmg section

Bodmg temperature surface In a previous article (7) we have proved that In a

c-component mrxture, maximum bollmg c-component azeotropes gwe nse to stable modes and mmlmum boding c-component azeotropes @ve nse to unstable nodes

The correspondence between the type of stationary pomt in the bothng temperature surface and the type of stablltty of the assocmted smgular point can be analyzed further by use of the followmg equation (see Doherty and Perkms[7]) for the determinant of the Jacobian matix at a multicomponent azeotrope of composition 2,

(37)

Assume a single ternary saddle does occur, then SX = 1, N, = 0, N2 = S2 = 0 hence eqn (34) becomes,

N,-S:=4 (35)

The left hand side of (35) has a rnaxlmum possible value of three So, we can state the general proposlfion

A ternary saddle point azeotrope cannot occur in a ternary mixture whtch exhtblts no bmary azeotropes

PhysIcally this IS a very reasonable result because a saddle pomt has four separatrrces, each of which requires a smgular pomt to tend towards (as g+ + CO or -m) In a ternary mixture with no bmary azeotropes there are only three singular points avdable (the ver- tices)

Example 2 Equation (34) becomes,

2N,-2&+2N2+N,-S:=S (361

Unhke the elementary case we cannot determme the nature of a pure component smgular point purely from the due&on of the arrows on the bmary faces Th~3 makes eqn (36) more unwieldy than Its counterpart, eqn (21)

Fwures IO(b)-(g) can be augmented by many more diagrams m which we allow for non-elementary vetices

where B 1s a positive number For a ternary nuxture this reduces to,

det J(z) = 8* det

where 8* IS a posttlve number Equation (38) tells us that saddle pomts m the bollmg

surface gave flse to smgular pomts which are saddle pomts and vice versa This equation also tells us that semldefinlte stationary pomts m the boding surface gwe nse to non-elementary smgular pomts and vice-versa

Because eqn (38) cannot mve any mformatton about the possible types of seml-defimte stationary pomts we were not able to answer the question of Just how “bad” these semtdefintte stationary points can be In general, semldefimte stationary pomts can be as comphcated as we choose, wtth multiple valleys and ndges convergmg on the stationary pomt

This question no longer remams unanswered We have proved that the only new smgular pomt Introduced by a zero elgenvalue LS the node-saddle shown m Fig 12(a)

Thus smgular pomt must @ve nse to an armchau hke stationary pomt m the bollmg temperature surface because the temperature always rises along a residue curve (see (7) Theorem Four) This proves

TI E 1110)

On the dynamtcs of dtstdlatlon processes-III 1411

B D T+ = 5 To (4s) (40)

I*01 tbl

Rg 13 Examples of non-elementary resrdue curve maps

trope to occur m a rmxture which exhlblts a bmary muumum bodmg azeotrope on each of the bmary faces

(IV) If a ternary saddle pomt azeotrope occurs, Its bollmg temperature can be bracketed Thus enables us to determme temperature mtervals m whch a ternary azeotrope cannot occur Thus goes some way towards answermg the problem described by Maruuchev[31 quoted in the mtroductlon, that, “the question of the exrstence of the ternary azeotrope at a gven tempera- ture remams open”

(v) Allowmg for non-elementary smgular pomts, It Is not possible for a smgle saddle point azeotrope to occur m a mixture which exhlblts no bmary azeotropes

(VI) The bollmg pomt surface can exhlblt only four types of stationary pomt maxlma, muuma, saddles and chatrs

Acknowledgement-One of us (M F D ) wshes to thank the Nattonal Science Foundatton for provldmg financial support m the form of a Research Imtlatlon Grant (Grant Number ENG 7% 05565)

NOTATION

A vector of parameters m eqn (2)

Theorem The only kmds of ternary azeotropes which can occur

are maxlmum bodmg, mmlmum bollmg. saddle pomt and armchau-hke azeotropes

The bolhng surface cannot be any more comphcated

B,(x) ball, centre x, radius r c number of components m a multicomponent

mrxture e, unit vector m the rth orthogonal dlrectlon e number of elhptlc sectors f vector functions

than the restictions Imposed by this theorem It IS worth notmg the tnvlal pomt that m bmary

nurtures the only kmd of seml-defimte stationary pomt which can occur IS an mflexlon

CONCLUSION

The dlstdlatlon eqn (2) place a physIcally meanmgful structure (I e tangent vector field) on ternary phase diagrams By recogmzmg that such structures are sublect to some very general mathematical laws It has been passable to obtam structural relationshIps between the azeotropes and pure components occumng tn ternary mixtures

These relationships gwe useful mformatlon about the dlstlllatlon behavlour of ternary mixtures and also allow us to obtam some general results concemmg the nature of ternary azeotropes These may be summarued as follows

(I) When all the smgular pomts are elementary rt IS not possible for a smgIe ternary azeotrope to occur m a mixture which exhlblts no bmary azeotropes

(u) When all the smgular pomts are elementary It 1s not possible for a smgle ternary nummum bolng azeo- trope to occur III a murture which exhibits a bmary maxlmum boding azeotrope on each of the bmary faces

(m) When all the smgular pomts are elementary it Is not possible for a smgle ternary maxnnum bofimg azeo-

N.%

mverse of f Image of an image of the set of cntical pomts Z genus (eqn 13) number of hyperbohc sectors mdex of smgular pomt mdex of a smgular pomt of type a Jacoblan matnx of first partA denvatlves of f compact boundaryless mamfold

number of nodes m the mangle of type Q number of node-saddles m the mangle of type a

P pressure P,O saturated vapour pressure of pure component I R gas constant

R” space of dlmenslon n S, number of saddles m the mangle of type a St number of pure component half saddles m the

-de T temperature

TaF bodmg temperature of pure component c x (1) abstract state vector, (2) vector of mole frac-

tions m the lrqmd phase y vector of mole fractions in the vapour phase 2 azeotrop~c composition

Z set of cntical pomts (1 e pomts such that J,(x) = 0)

1412 M F DonmtnandJ D PERKINS

Greek symbols Yi achvity coefficient of component I Ai elgenvalue @ open rtght angled trtangle with unit side (the open

composition h-tangle) 13@ the boundary of @

6 the closure of @ R open set m R”

an the boundary of n fi the closure of Q 2‘ dlmensroniess tie 9

19* posrttve numbers x Euler number

Mathemattcal symbols [ ] closed interval E contamed m !z not contamed m

sgn sign

Superscripts m mfirute dllutron 0 pure component

AZ azeotroprc value

Subscripts I, J dummy subscripts

x’ keep the vanables xl, x2 stant

q-t, xj+b xc-l con-

PI

r21

f31

r41

r51

[61

r71

181

191

[toI

[ttl

WI

u31

1141

WI

WI

[I71

REFERENCES

Herdemann R A and Mandhane J M , Chem Engng SCI 1975 38 425 Marmrchev A N , J Appl Chem USSR (Eng Tmns ) 1971 44 2399 Maruuchev A N , J Appi Chem USSR (Eng Tmns ) 1972 43 2602 Susarev M P and Totia A M , Russ J Phys Chem (Eng Trans) 1974 48 1584 Mamuchev A N and Vmtchenko 1 G , J ADD/ Chem _ _ USSR (Eng Tmns ) I%9 42 525 Doherty M F and Perkms J D. Chem Ennnp Scr 1977 32 1112 -

-_

Doherty M F and Perkms J D, Chem Engng Scr 1978 33 281 Doherty M F and Perkms J D, Chem Engng Scr 1978 33 569 Temme N h4 (Ed ), Noniuteur Analysrs, Vol 1 Mathema- ttsch Centrum, Amsterdam 1976 Lefschetz S , D$etzntIal Equatwns Geometnc Theory, 2nd Edn Intersctence. New York 1957 Gavalas G R , No&near Dtffenznturl Equutrons of Chem- uxlly Reucturg Systems Sormger-Verlag, New York l%g Courant R and Robbms H , In The World of Mathematrcs (Edged by Newman J R 1. DD 581-599 Stmon and Schus- ter. New -York. 1956 .- _ _ DoCarmo M P , hfferentuzl Geometry of Curves and Sur- faces Prentrce-Hall;Englewood Cl& Nkw Jersey 1976 Gudlemm V and Pollack A. LWerentral Tooolo~v Pren- tree-Hall. Englewood Cldfs, New jersey 1974 _ -_ Mdnor, J W , Topology fmm the D#erentmble Kewpornt The Unrverstty Press of Vugmta, Charlottesvdle 1976 Wade J C and Taylor Z L , I Chem Engng aOtu 1973 18 424 Zemrke J . Chemrcai Phase Theory Kluwer, Antwerp, 1956

[W

1191

1201

Et1

[221

1231

I241

WI

WI

1271

Horsley L H , Azeottvprc Data III Amerrcan Chemical Socrety, Wasbmgton. D C 1973 Sansone G and Conb R , Nonlmear Dtff-erenttal Equotrons Pergamon Press, Oxford 1%4 Ewe11 R H and Welch L M , Znd &gng Chem 1945 37 1224 Remders W and DeMmler C H , Ret Trau Chrm 1940 59 207 Remders W and DeMmJer C H , Ret True Chrm 1940 59 369 Remders W and DeMinJer C H , Ret Tmw Chum 1940 59 392 Wilson R Q , Mink W H , Munger H P and Clegg J W, AIChEJ.19552220 Rowlmson J S , Lrqurds and bqurd Mxtures, 2nd Edn Buttenvorths, London 1%9 Horsley L H , Azeotroprc Data [II Advances m Chemistry Senes 116 Amencan Chemical Soctety. Washmgton, D C 1973 Benedtct M and Rubm L C , Natwnal Petroleum News 1945 37 R729

AFP-I

The drsttllatton equattons for a c-component mtxture are,

dx

dz=x-y

Wdhout loss of generabty we can study the properttes of these equattons at any vertex by studymg them at the ortgm (see Doherty and Perkms[8], Appendtx I)

Lmeanzmg (Al) about the ongm gtves.

dx I-Y,,

Clf=

1 -Y22

0 1

0

- Yc-

1 X

(A21

-1 r--l J where Y,, = (~Y&WP = =O The off-dtagonal terms are zero because y, = 0 when x, = 0 for all values of x,, I# I hence w,ia+ x =a = 0 for all J# I (see Frg Al and eqn (32) m Ref

[81) The Jacobtan matnx m (A2) IS diagonal so Its normalized ergen

vectors are the orthonormal duecttons e,, ez. e3 e,_, It IS well known that the soluttons of a system of lmear ordmary ddferen- teal equatrons exhibit the properttes,

(I) If the ortgm IS a node, the tralectones approach tangenttalty along an ergendtrectton (see Appendtx II)

(II) If the ortgm IS a saddle, the traJectorres become asymptottc to the etgenduecttons This lustrfles Figs 7(c) and (d) and proves that these are the only elementary pure component smgular pomts

Of course, the above reasonmg fads tf y,, = 1 for any I Phystcally, the term y,, represents the mrttal slope of the yi vs

xl diagram for the bmary submrxture of components l and c (see Fig Al)

TJus slope can be wrttten m terms of other thermodynamtc quantrttes as follows Assume an eqmhbnum relatton of the form,

PYr = PPx,r, (A3)

DdIerentratmg with respect to x, and letting x, = x2 = X,_, = 0 gtves

( ) ay, = rPC’( Tic ) ax, P I -0 P (A4)

where yi” IS the mtimte ddutron acttvity coe&tent of component I and P,‘(Tb,) IS the saturated vapour pressure of pure I at the

OF I All0 3 When a tangentml azeotrope behveen components k and r (rf c) occurs at vertex cI then

(3) ( > app a-% P II

*3 gwes

Fig Al Vapour-lrqtud composrtlon surface, y, vs x, and xt for a a non-azeotroplc ternary mixture Pco(ekh (ek)-pro(ek)Yr(ek) r= I

c-l

bodmg temperature of pure component c The r&t hand side of (A4) IS equal to umty when the binary mixture of components L (A13)

and c exhlbrt a tangenhal azeotrope at the orlgm, m whmh case We need to establish the condltlons under which the binary mixture of k and r exhlblts a tangentml azeotrope at vertex ek

cl W) The basic condltlon which we must work from IS

We can show this correspondence as follows Usmg the same 0 (A14) procedure described m Appendix I of Doherty and Perkms 181 we obtam where z denotes a movement m the k - r face, I e such that

dT ( > ax, = p - P3 L ) Y,“(O)

++c) W) q + x, = I (AIS) P x-0

From (A6) we see

Usmg the chain rule on (A14) gives

dT ( > *, ~ a*, p x a~ ( >

ax,=, ax, p I az (Al6)

On the dynanucs of dlstdlatlon processes-III 1413

PP(e&P(ek) = l P W 1)

and hence by (A9) we get A. = 0 We prove statement (Al 1) below Summmg eqn (A3) over all I

gwes

C-l PCOYA1 -11 -x2 --XC_,)+ c P#%,y, = P (A13

,-I

DtlTerentlatrng (Al2) with respect to x, r = l c - 1, taking the hnut as xk * 1 and remembermg,

(1)

(2)

Yk(ek) = 1

r,-.L ax, #?.+O lim a

I > r=l c-l

Between (AlS) and (Al6) we find the requued condltlon,

This corresoondence extends to all vertmes but ts more dltlicult to ex&act for those other than the oruun (Al7)

The Jacobian matnx for the vertex elr IS gwen by (see Ref WI)

1 Between (A13) and (A17) we find,

0 P%k )%*(e&) = Pk%k )Yk(ek) (‘418)

J(ek I= The rrght hand side of (A18) IS equal to the total pressure, P. and so we obtam

l_J% py:-det)

I P,“(e,)vr%t) = ,

P (Al9)

(A81 Equation (Al9) IS the condttlon for a tangential azeotrope to

The elgenvalues are @ven by occur between components k and r at vertex et This completes what we set out to prove

A = , _ P%kh%k) P 1=1 C-l I# k (A91

In summary, we have shown that one zero ergenvalue IS Introduced for every tangentml azeotrope occumng at a given vertex

Ak= -~~(~~)(~)p (AlO) .% -en APPENDIXU

When a tangentml azeotrope between components k and c Shape of the rexdue cumes in the vwmty of pure component occurs at vertex ek then (AIO) shows that Ak = 0 venkces

1414 M F L&WFXY and J D &SKINS

From eqns (A2) and (A4) we find that the ergenvalues of the Kk=(er ) = rc”(ek )PP( TM)

bnearlzed equations about the 0~ are, P

A, = 1 - K,_(O) 1= 1 C-l (A20) We wdl wnte eqns (A20) and (A22) as,

where K,-(O) IS the mfimte dllutlon K-value of component I evaluated at the ongm, A,=]-KI”

(‘424)

(A=)

Y,-(o)pP(Tbc) _ aY1 K-(O) = p -z ( ) I=1 c-l It bemg understood that the Kp”‘s are evaluated In the proper way P r=0 at the vertex of Interest

(A=) In accordance with the theory of lmear differential equations we have

From eqns (A9). (AlO) and (Al3) we find that the elgenvalues of (a) The vertex IS a stable node d K,“C 1, I = 1 c-l The the lmeanzed equahons about vertex q are, residue curves are tangent to the hne connectmg the vertex about

A, = 1 - ZCp”(ek) (A221 which we have lmeamed with vertex V, such that A. = max {A,} (I e A. has the smallest absolute value)

c-l where K;“(et) IS the mfnute dllutlon K-value of component I

(b) The vertex IS an unstable node d K,- > 1. d= 1

evaluated at the vertex ek, The restdue curves are tangent to the bne connectmg the vertex about which we have bneanzed w&h vertex 0. such that A. = mm i&l

s=l c-l r#k (c) The vertex IS a saddle under any other cvcumstances

(A=) (except the particular case Kjm = 1 for some J This case has already been dlscussed m Appeadlx I)